From e056f7af55734103c4e2b729648fbb22f3b4f0eb Mon Sep 17 00:00:00 2001
From: Per Alexandersson <per.w.alexandersson@gmail.com>
Date: Wed, 21 Oct 2026 11:54:43 +0000
Subject: [PATCH] Catalog: share the symmetricfunctions.com bibliography and
 labels

Catalog pages can now cite the SymCat bibliography and link SymCat
statement labels:

- `scripts/symcat_sync.py` snapshots the public `site-labels.json` and
  the selected entries of `bibliography.json` into `website/symcat/`.
- Page metadata takes `references = [...]` and declaration-level
  `symcat = "label"`; prose takes `[@Key, locator]` citations and
  `sf:label` links. Pages get a generated reference list with BibTeX.
- The manifest lists each page's declarations with their SymCat labels.
- `build_challenge_pages.py` gains `--base-path` and `--site-origin`.

170 page/key citations on 81 pages now use 123 shared keys. Lean files
change only inside module docstrings.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
---
 .github/workflows/lean_action_ci.yml          |    1 +
 .../Challenges/AissenSchoenbergWhitney.lean   |   13 +-
 RealRooted/Challenges/Bezoutian.lean          |   13 +-
 RealRooted/Challenges/BigDescents321.lean     |   12 +-
 .../Challenges/BinaryRunTransformation.lean   |   20 +-
 RealRooted/Challenges/BorceaBranden.lean      |   15 +-
 .../BorceaBrandenClassification.lean          |   12 +-
 RealRooted/Challenges/BorosMoll.lean          |   27 +-
 RealRooted/Challenges/BrandenETransform.lean  |   11 +-
 RealRooted/Challenges/BrandenLeite.lean       |   12 +-
 RealRooted/Challenges/BrandenSolus.lean       |   14 +-
 RealRooted/Challenges/BrandenVecchi.lean      |   11 +-
 RealRooted/Challenges/CauchyInterlacing.lean  |   12 +-
 .../ChordalClawFreeAcyclicSinks.lean          |    2 +-
 RealRooted/Challenges/ChudnovskySeymour.lean  |   14 +-
 .../Challenges/ClawFreeOrientationSinks.lean  |    2 +-
 .../Challenges/CoefficientDominance.lean      |    9 +-
 RealRooted/Challenges/CommonInterleaver.lean  |   20 +-
 RealRooted/Challenges/CourantFischer.lean     |   12 +-
 RealRooted/Challenges/DecoEulerian.lean       |   11 +-
 .../Challenges/DerivativeInterlacing.lean     |   21 +-
 RealRooted/Challenges/Descartes.lean          |    2 +-
 RealRooted/Challenges/EdreiThoma.lean         |   16 +-
 RealRooted/Challenges/EndpointDerivative.lean |   13 +-
 RealRooted/Challenges/EulerBidiagonal.lean    |   22 +-
 RealRooted/Challenges/Eulerian.lean           |   21 +-
 RealRooted/Challenges/EulerianVariations.lean |    6 +-
 .../Challenges/FPolynomialRecurrences.lean    |   12 +-
 .../Challenges/FactorialCompression.lean      |    8 +-
 RealRooted/Challenges/Favard.lean             |   12 +-
 .../Challenges/FiniteFreeConvolutions.lean    |   20 +-
 RealRooted/Challenges/FiniteMemoryDefect.lean |   13 +-
 RealRooted/Challenges/GammaTransform.lean     |   20 +-
 .../Challenges/GarloffWagnerHurwitz.lean      |   16 +-
 .../Challenges/GeneralizedSnakePosets.lean    |   23 +-
 RealRooted/Challenges/GraceApolarity.lean     |   39 +-
 RealRooted/Challenges/Hadamard.lean           |   54 +-
 RealRooted/Challenges/HarperCLT.lean          |   23 +-
 RealRooted/Challenges/HeilmannLieb.lean       |   28 +-
 .../Challenges/HermiteBiehlerHurwitz.lean     |   24 +-
 RealRooted/Challenges/HermitePolynomials.lean |    6 +-
 RealRooted/Challenges/HermitePoulain.lean     |    6 +-
 RealRooted/Challenges/HosterStump.lean        |   12 +-
 RealRooted/Challenges/Hutchinson.lean         |   14 +-
 RealRooted/Challenges/Interlacing.lean        |    9 +-
 RealRooted/Challenges/InterlacingClosure.lean |   22 +-
 .../Challenges/IteratedDerivativeShift.lean   |   11 +-
 RealRooted/Challenges/JacobiDeformation.lean  |   18 +-
 RealRooted/Challenges/JacobiPolynomials.lean  |   14 +-
 .../Challenges/JacobiStirlingDescent.lean     |   14 +-
 .../Challenges/JacobiStirlingNumbers.lean     |   16 +-
 .../Challenges/KostovEigenpolynomials.lean    |   12 +-
 RealRooted/Challenges/Kurtz.lean              |   11 +-
 RealRooted/Challenges/LGV.lean                |   12 +-
 .../Challenges/LaguerrePolarDerivative.lean   |    6 +-
 RealRooted/Challenges/LaguerrePolya.lean      |   12 +-
 .../Challenges/LaguerrePolynomials.lean       |    7 +-
 RealRooted/Challenges/LeakeRyder.lean         |   17 +-
 RealRooted/Challenges/LeeYang.lean            |   18 +-
 RealRooted/Challenges/Liggett.lean            |   10 +-
 RealRooted/Challenges/LiuOppositeSigns.lean   |    9 +-
 RealRooted/Challenges/MaWangLiuWang.lean      |   19 +-
 RealRooted/Challenges/MatrixInterlacing.lean  |   14 +-
 .../Challenges/MonomialChainOperator.lean     |    7 +-
 .../Challenges/MonotoneColumnPermanent.lean   |   19 +-
 RealRooted/Challenges/MorrisFlow.lean         |   18 +-
 RealRooted/Challenges/MultiplierSequence.lean |    8 +-
 .../Challenges/MultisetEulerianNarayana.lean  |   23 +-
 .../Challenges/MultivariateObreschkoff.lean   |   21 +-
 RealRooted/Challenges/Narayana.lean           |   37 +-
 RealRooted/Challenges/NewtonDefectRoots.lean  |   13 +-
 RealRooted/Challenges/Nijenhuis.lean          |    9 +-
 RealRooted/Challenges/Obreschkoff.lean        |   16 +-
 RealRooted/Challenges/OperatorPreservers.lean |   17 +-
 RealRooted/Challenges/ParkingFunctions.lean   |   11 +-
 RealRooted/Challenges/PathPowerCliques.lean   |   11 +-
 RealRooted/Challenges/PeakPolynomials.lean    |    2 +-
 RealRooted/Challenges/PerronFrobenius.lean    |   13 +-
 RealRooted/Challenges/ProperPositionPF.lean   |   16 +-
 RealRooted/Challenges/RealStability.lean      |   63 +-
 RealRooted/Challenges/ReciprocalShift.lean    |    7 +-
 .../Challenges/RecurrenceRootWindows.lean     |   12 +-
 RealRooted/Challenges/RootAmplitude.lean      |    9 +-
 RealRooted/Challenges/SchurCohn.lean          |   12 +-
 .../Challenges/SeparablePermutations.lean     |   23 +-
 .../Challenges/SmallInterlacingMatrices.lean  |   13 +-
 RealRooted/Challenges/Sturm.lean              |   23 +-
 .../Challenges/SturmSequenceFamilies.lean     |   39 +-
 RealRooted/Challenges/SzaszGrowth.lean        |   14 +-
 RealRooted/Challenges/ThresholdMatrices.lean  |   25 +-
 RealRooted/Challenges/ToricContribution.lean  |   13 +-
 ...TotallyNonnegativeHadamardObstruction.lean |    9 +-
 .../TotallyNonnegativeMatrices.lean           |   30 +-
 .../UniformMatroidHodgePoincare.lean          |   24 +-
 .../Challenges/UnivariatePoissonStar.lean     |    8 +-
 RealRooted/Challenges/VanDerWaerden.lean      |   22 +-
 RealRooted/Challenges/VeroneseSections.lean   |   13 +-
 RealRooted/Challenges/Wagner.lean             |   11 +-
 RealRooted/Challenges/WangYeh.lean            |   17 +-
 RealRooted/Challenges/Wronskian.lean          |    9 +-
 scripts/build_challenge_pages.py              |   30 +-
 scripts/challenge_catalog.py                  |  464 ++-
 scripts/symcat_sync.py                        |  178 +
 scripts/test_challenge_catalog.py             |  164 +
 scripts/test_symcat_sync.py                   |  117 +
 website/README.md                             |   21 +
 website/assets/site.css                       |   29 +-
 website/symcat/bibliography.json              | 3519 +++++++++++++++++
 website/symcat/labels.json                    |    1 +
 109 files changed, 5265 insertions(+), 820 deletions(-)
 create mode 100644 scripts/symcat_sync.py
 create mode 100644 scripts/test_symcat_sync.py
 create mode 100644 website/symcat/bibliography.json
 create mode 100644 website/symcat/labels.json

diff --git a/.github/workflows/lean_action_ci.yml b/.github/workflows/lean_action_ci.yml
index aaf8f057f..370e8b5af 100644
--- a/.github/workflows/lean_action_ci.yml
+++ b/.github/workflows/lean_action_ci.yml
@@ -25,6 +25,7 @@ jobs:
       - name: Check catalog sources and renderer
         run: |
           PYTHONDONTWRITEBYTECODE=1 python3 scripts/test_challenge_catalog.py
+          PYTHONDONTWRITEBYTECODE=1 python3 scripts/test_symcat_sync.py
           PYTHONDONTWRITEBYTECODE=1 python3 scripts/test_challenge_catalog_audit.py
           PYTHONDONTWRITEBYTECODE=1 python3 scripts/build_challenge_pages.py --check
           PYTHONDONTWRITEBYTECODE=1 python3 scripts/audit_challenge_catalog.py --check
diff --git a/RealRooted/Challenges/AissenSchoenbergWhitney.lean b/RealRooted/Challenges/AissenSchoenbergWhitney.lean
index 489d948f2..8d808dd10 100644
--- a/RealRooted/Challenges/AissenSchoenbergWhitney.lean
+++ b/RealRooted/Challenges/AissenSchoenbergWhitney.lean
@@ -11,6 +11,9 @@ version = 1
 section = "theorems"
 slug = "aissen-schoenberg-whitney"
 formalized = "2026-10-05"
+references = [
+  "AissenSchoenbergWhitney1952",
+]
 authors = ["Aissen", "Schoenberg", "Whitney"]
 years = [1952]
 
@@ -60,6 +63,8 @@ label = "Karlin: zeros of PF polynomials avoid a sector around the positive axis
 <!-- realrooted-catalog-content -->
 # Aissen–Schoenberg–Whitney
 
+[@AissenSchoenbergWhitney1952]
+
 A real sequence $(a_k)_{k \geq 0}$ is a Pólya frequency sequence if its
 lower-triangular Toeplitz matrix, whose $(i, j)$ entry is $a_{i-j}$ for
 $j \leq i$ and $0$ otherwise, is totally nonnegative. The coefficient sequence
@@ -82,14 +87,12 @@ totally nonnegative matrices is totally nonnegative.
 
 ## References
 
-M. Aissen, I. J. Schoenberg, and A. M. Whitney, “On the generating functions
-of totally positive sequences. I,” *Journal d’Analyse Mathématique* 2 (1952),
-93–103, [doi:10.1007/BF02786970](https://doi.org/10.1007/BF02786970). See the
-[Pólya-frequency overview on symmetricfunctions.com](https://www.symmetricfunctions.com/polyaFrequency.htm#aissenSchoenbergWhitney).
+See [@AissenSchoenbergWhitney1952] and the
+[Pólya-frequency overview on symmetricfunctions.com](sf:aissenSchoenbergWhitney).
 <!-- /realrooted-catalog-content -->
 
 Human statement:
-https://www.symmetricfunctions.com/polyaFrequency.htm#aissenSchoenbergWhitney
+sf:aissenSchoenbergWhitney
 
 Original publication: M. Aissen, I. J. Schoenberg, and A. M. Whitney,
 "On the generating functions of totally positive sequences. I",
diff --git a/RealRooted/Challenges/Bezoutian.lean b/RealRooted/Challenges/Bezoutian.lean
index 2049e85c7..20eb01e4f 100644
--- a/RealRooted/Challenges/Bezoutian.lean
+++ b/RealRooted/Challenges/Bezoutian.lean
@@ -9,6 +9,9 @@ version = 1
 section = "theorems"
 slug = "bezoutian"
 formalized = "2026-10-06"
+references = [
+  "Fisk2006x",
+]
 authors = ["Hermite", "Krein", "Naimark", "Fisk"]
 years = [1856, 1936, 2006]
 
@@ -64,7 +67,8 @@ Positive definiteness alone already forces both polynomials to split over
 $\mathbb R$, and it makes the Wronskian $q'p - qp'$ positive on all of
 $\mathbb R$.
 
-**Theorem** (successor degree). Let $f$ have degree $d+1$ and $g$ degree $d$,
+**Theorem** (successor degree). [@Fisk2006x, Cor. 9.145] Let $f$ have degree
+$d+1$ and $g$ degree $d$,
 both with positive leading coefficients. Then $g$ interlaces $f$ and the two
 polynomials have no common zero, that is
 $$
@@ -89,10 +93,9 @@ C. Hermite, “Sur le nombre des racines d'une équation algébrique comprise
 entre des limites données,” *J. Reine Angew. Math.* 52 (1856), 39–51;
 M. G. Krein and M. A. Naimark, “The method of symmetric and Hermitian forms in
 the theory of the separation of the roots of algebraic equations” (1936),
-English translation in *Linear and Multilinear Algebra* 10 (1981), 265–308;
-S. Fisk, *Polynomials, roots, and interlacing*, arXiv:math/0612833, Cor. 9.145.
-See the
-[contextual statement on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedInterlacing.htm#bezoutMatrix).
+English translation in *Linear and Multilinear Algebra* 10 (1981), 265–308.
+The successor-degree criterion is [@Fisk2006x, Cor. 9.145]. See the
+[contextual statement on symmetricfunctions.com](sf:bezoutMatrix).
 <!-- /realrooted-catalog-content -->
 
 This module exposes the checked same-degree and successor-degree Bézout
diff --git a/RealRooted/Challenges/BigDescents321.lean b/RealRooted/Challenges/BigDescents321.lean
index fc21ec065..dfc615d38 100644
--- a/RealRooted/Challenges/BigDescents321.lean
+++ b/RealRooted/Challenges/BigDescents321.lean
@@ -8,6 +8,9 @@ version = 1
 section = "theorems"
 slug = "big-descents-321"
 formalized = "2026-10-06"
+references = [
+  "ElizaldeRiveraZhuang2026",
+]
 authors = ["Elizalde", "Rivera", "Zhuang", "Alexandersson"]
 years = [2024, 2026]
 
@@ -55,10 +58,11 @@ polynomials.
 simple and strictly negative, $M_{n-2}$ interlaces $A_n$, and the two polynomials have no common
 root.
 
-This settles the case $\Pi = \{321\}$ of Conjecture 4.4 of Elizalde, Rivera and Zhuang. The Lean
+This settles the case $\Pi = \{321\}$ of Conjecture 4.4 of Elizalde, Rivera and Zhuang
+[@ElizaldeRiveraZhuang2026, Conj. 4.4]. The Lean
 statements are about the recurrence-defined polynomials; the enumeration of permutations is the
 combinatorial interpretation and is not formalized. See also
-[big descents on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedWords.htm#bigDescent).
+[big descents on symmetricfunctions.com](sf:bigDescent).
 
 ## Proof idea
 
@@ -81,9 +85,7 @@ The cases $n \le 24$ are exact interlacing and Bezout certificates.
 
 ## References
 
-S. Elizalde, J. Rivera, Jr., and Y. Zhuang, [“Counting pattern-avoiding permutations by big
-descents,”](https://arxiv.org/abs/2408.15111) *Annals of Combinatorics* (2026),
-[doi:10.1007/s00026-026-00805-1](https://doi.org/10.1007/s00026-026-00805-1).
+See [@ElizaldeRiveraZhuang2026].
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in
diff --git a/RealRooted/Challenges/BinaryRunTransformation.lean b/RealRooted/Challenges/BinaryRunTransformation.lean
index a965e3d7e..4de547e7b 100644
--- a/RealRooted/Challenges/BinaryRunTransformation.lean
+++ b/RealRooted/Challenges/BinaryRunTransformation.lean
@@ -12,6 +12,10 @@ version = 1
 section = "theorems"
 slug = "binary-run-transformation"
 formalized = "2026-09-28"
+references = [
+  "BorceaBranden2009I",
+  "MaoWang2026x",
+]
 
 [[definitions]]
 name = "RealRooted.binaryRunPolynomial"
@@ -98,7 +102,8 @@ sends $X^m$, for $1 \leq m \leq n$, to
 
 $$\frac{1}{\binom{n}{m}} \sum_k \binom{m-1}{k-1}\binom{n+1-m}{k} X^k.$$
 
-If the input has degree at most $n$, nonnegative coefficients, and only real
+By the algebraic-symbol method [@BorceaBranden2009I], if the input has degree
+at most $n$, nonnegative coefficients, and only real
 nonpositive zeros, then its image has the same properties. If the input has
 positive constant coefficient, every zero of the image is strictly negative.
 
@@ -110,7 +115,8 @@ $$T_n(X^m) = \frac{\binom{n}{m}}{m+1}\, \mathtt{binaryRunPolynomial}\ n\ m.$$
 Equivalently, it is `binaryRunTransform n` after the standard finite
 Narayana/Schur–Szegő diagonal multiplier. Combined with preservation by that
 multiplier, this preservation theorem yields the corresponding
-[Narayana transformation theorem](/RealRooted/families/narayana/); that
+[Narayana transformation theorem](/RealRooted/families/narayana/)
+[@MaoWang2026x]; that
 deduction is cited, and Lean does not connect `binaryRunTransform` to
 `narayanaTransform`.
 
@@ -157,14 +163,8 @@ prevents collisions throughout the deformation.
 
 ## References
 
-The stability-preserving contraction uses the algebraic-symbol machinery of
-J. Borcea and P. Brändén, [“The Lee–Yang and Pólya–Schur programs. I. Linear
-operators preserving stability,”](https://arxiv.org/abs/0809.0401)
-*Inventiones Mathematicae* 177 (2009), 541–569,
-[doi:10.1007/s00222-009-0189-3](https://doi.org/10.1007/s00222-009-0189-3).
-For the related transform, see
-J. Mao and L. Wang, [“The Narayana transformation,”](https://arxiv.org/abs/2607.01572)
-arXiv:2607.01572 (2026).
+The stability-preserving contraction uses the algebraic-symbol machinery
+[@BorceaBranden2009I]. For the related Narayana transform, see [@MaoWang2026x].
 <!-- /realrooted-catalog-content -->
 
 This module exposes the strictly negative root corollary of the checked
diff --git a/RealRooted/Challenges/BorceaBranden.lean b/RealRooted/Challenges/BorceaBranden.lean
index 8ac935994..8fd0b098c 100644
--- a/RealRooted/Challenges/BorceaBranden.lean
+++ b/RealRooted/Challenges/BorceaBranden.lean
@@ -13,6 +13,9 @@ version = 1
 section = "theorems"
 slug = "borcea-branden"
 formalized = "2026-08-09"
+references = [
+  "BorceaBranden2009I",
+]
 authors = ["Borcea", "Brändén"]
 years = [2009]
 
@@ -72,7 +75,8 @@ label = "Poisson left multiplication preserves real stability"
 <!-- realrooted-catalog-content -->
 # Borcea–Brändén finite-symbol theorems
 
-A linear operator on a finite multidegree box preserves stability precisely in
+A linear operator on a finite multidegree box preserves stability precisely
+as described by the finite-symbol classification [@BorceaBranden2009I]: in
 the rank-one or stable-symbol cases. The real univariate theorem gives the
 corresponding positive-symbol criterion for bounded-degree polynomials.
 
@@ -96,15 +100,12 @@ operator is $f(z(1-w), v(1-w))$.
 
 ## References
 
-J. Borcea and P. Brändén, “The Lee–Yang and Pólya–Schur programs. I. Linear
-operators preserving stability,” *Inventiones Mathematicae* 177 (2009),
-541–569, [doi:10.1007/s00222-009-0189-3](https://doi.org/10.1007/s00222-009-0189-3).
-See the
-[finite-symbol discussion on symmetricfunctions.com](https://www.symmetricfunctions.com/stablePolynomials.htm#borceaBrandenFiniteSymbol).
+See [@BorceaBranden2009I] and the
+[finite-symbol discussion on symmetricfunctions.com](sf:borceaBrandenFiniteSymbol).
 <!-- /realrooted-catalog-content -->
 
 Human statement:
-https://www.symmetricfunctions.com/stablePolynomials.htm#borceaBrandenFiniteSymbol
+sf:borceaBrandenFiniteSymbol
 
 Original reference: J. Borcea and P. Branden, "The Lee-Yang and Polya-Schur
 programs. I. Linear operators preserving stability", Invent. Math. 177 (2009),
diff --git a/RealRooted/Challenges/BorceaBrandenClassification.lean b/RealRooted/Challenges/BorceaBrandenClassification.lean
index 6180f7e49..fd36a4a37 100644
--- a/RealRooted/Challenges/BorceaBrandenClassification.lean
+++ b/RealRooted/Challenges/BorceaBrandenClassification.lean
@@ -12,6 +12,9 @@ version = 1
 section = "theorems"
 slug = "borcea-branden-classification"
 formalized = "2026-10-13"
+references = [
+  "BorceaBranden2009I",
+]
 authors = ["Borcea", "Brändén"]
 years = [2009]
 
@@ -144,6 +147,8 @@ limits of complex stable-or-zero polynomials.
 `IsRealStableLimit` consists of entire real-coefficient series that are
 locally uniform limits of real stable polynomials.
 
+The theorem numbers below refer to [@BorceaBranden2009I].
+
 Theorem 1.2 is
 `preservesRealStabilityOnDegreeBox_iff_rankTwo_or_symbol_or_negRight`: a
 real-linear operator on a coordinatewise degree box preserves real stability
@@ -178,12 +183,7 @@ using specialization and partial differentiation.
 
 ## References
 
-J. Borcea and P. Brändén,
-[“The Lee–Yang and Pólya–Schur programs. I. Linear operators preserving
-stability,”](https://arxiv.org/abs/0809.0401) *Invent. Math.* 177
-(2009), 541–569,
-[doi:10.1007/s00222-009-0189-3](https://doi.org/10.1007/s00222-009-0189-3),
-[arXiv:0809.0401](https://arxiv.org/abs/0809.0401).
+See [@BorceaBranden2009I].
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/BorosMoll.lean b/RealRooted/Challenges/BorosMoll.lean
index 7ffe2709f..79e534474 100644
--- a/RealRooted/Challenges/BorosMoll.lean
+++ b/RealRooted/Challenges/BorosMoll.lean
@@ -9,6 +9,11 @@ version = 1
 section = "theorems"
 slug = "boros-moll-log-concavity-transform"
 formalized = "2026-10-08"
+references = [
+  "XieZhang2026InfiniteLogConcavity",
+  "Branden2011b",
+  "BorosMoll1999",
+]
 authors = ["Xie", "Zhang"]
 years = [2026]
 
@@ -68,20 +73,23 @@ $M_n(x) = \sum_i L(d(n))_i x^i$, and let
 $N_n(x) = \sum_{i=0}^n \frac{1}{i+1} \binom{n}{i} \binom{n+1}{i} x^i$ be the
 Narayana polynomial.
 
-**Theorem** (Xie–Zhang). For $n \geq 1$, $M_n$ has $n$ simple negative zeros, and they
+**Theorem** (Xie–Zhang). [@XieZhang2026InfiniteLogConcavity] For $n \geq 1$,
+$M_n$ has $n$ simple negative zeros, and they
 strictly interlace the zeros of $N_n$, those of $M_n$ being the leftmost.
 
-**Theorem** (Brändén). If $p$ has nonnegative coefficients and only real, nonpositive
-zeros, then so does its log-concavity transform. Hence every Pólya-frequency polynomial is
+**Theorem** (Brändén). [@Branden2011b] If $p$ has nonnegative coefficients and
+only real, nonpositive zeros, then so does its log-concavity transform. Hence every
+Pólya-frequency polynomial is
 *infinitely log-concave*: all iterates of the transform have nonnegative coefficients.
 
 Since $M_n$ is the transform of the row $d(n)$ and is Pólya-frequency, the Boros–Moll rows
-are infinitely log-concave, as conjectured by Boros and Moll. All three statements are
+are infinitely log-concave, as conjectured by Boros and Moll [@BorosMoll1999].
+All three statements are
 formalized. The formal proof of Brändén's theorem is not Brändén's: it uses moment
 functionals that do not vanish on polynomials whose zeros lie in a half-plane, together
 with Laguerre-type approximations.
 
-See also [infinite log-concavity on symmetricfunctions.com](https://www.symmetricfunctions.com/realRooted.htm#infinitelyLogConcave).
+See also [infinite log-concavity on symmetricfunctions.com](sf:infinitelyLogConcave).
 
 ## Proof idea
 
@@ -93,13 +101,8 @@ certificates evaluated in the kernel. The formal proof follows the paper.
 
 ## References
 
-P. Brändén, “Iterated sequences and the geometry of zeros,” *Journal für die reine und
-angewandte Mathematik* 658 (2011), 115–131,
-[doi:10.1515/crelle.2011.063](https://doi.org/10.1515/crelle.2011.063); M. H. Y. Xie and P. B. Zhang, “Infinite
-log-concavity of the Boros–Moll sequences,”
-arXiv:2609.20653 (2026); G. Boros and V. H. Moll, “An integral hidden in Gradshteyn and
-Ryzhik,” *J. Comput. Appl. Math.* 106 (1999), 361–368,
-[doi:10.1016/S0377-0427(99)00081-3](https://doi.org/10.1016/S0377-0427(99)00081-3).
+See [@Branden2011b; @BorosMoll1999]. For the infinite log-concavity result,
+see [@XieZhang2026InfiniteLogConcavity].
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in
diff --git a/RealRooted/Challenges/BrandenETransform.lean b/RealRooted/Challenges/BrandenETransform.lean
index 5c6ab40c4..6a29b22fa 100644
--- a/RealRooted/Challenges/BrandenETransform.lean
+++ b/RealRooted/Challenges/BrandenETransform.lean
@@ -9,6 +9,9 @@ version = 1
 section = "theorems"
 slug = "branden-e-transform"
 formalized = "2026-09-14"
+references = [
+  "Branden2006",
+]
 authors = ["Brändén"]
 years = [2006]
 
@@ -68,7 +71,7 @@ $E(x^n) = F_n(x)$. Equivalently,
 $E\bigl(x(x - 1) \cdots (x - n + 1)\bigr) = n!\, x^n$, that is,
 $E\binom{x}{n} = x^n$.
 
-**Theorem (Brändén).** Let $p$ be a real polynomial whose roots are all real
+**Theorem (Brändén).** [@Branden2006] Let $p$ be a real polynomial whose roots are all real
 and lie in $[-1, 0]$.
 
 - If $p$ has positive leading coefficient, then $E(p)$ is a PF polynomial: it
@@ -92,10 +95,8 @@ combination of an interlacing sequence is real-rooted.
 
 ## References
 
-P. Brändén, “On linear transformations preserving the Pólya frequency
-property,” *Transactions of the American Mathematical Society* 358 (2006),
-3697–3716, [doi:10.1090/S0002-9947-06-03856-6](https://doi.org/10.1090/S0002-9947-06-03856-6).
-The ordered Bell numbers are [OEIS A000670](https://oeis.org/A000670).
+See [@Branden2006]. The ordered Bell numbers are
+[OEIS A000670](https://oeis.org/A000670).
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade. The proofs live in
diff --git a/RealRooted/Challenges/BrandenLeite.lean b/RealRooted/Challenges/BrandenLeite.lean
index edc38ec6b..4c8e3fdc2 100644
--- a/RealRooted/Challenges/BrandenLeite.lean
+++ b/RealRooted/Challenges/BrandenLeite.lean
@@ -15,6 +15,9 @@ version = 1
 section = "theorems"
 slug = "branden-leite"
 formalized = "2026-09-24"
+references = [
+  "BrandenLeite2024x",
+]
 authors = ["Brändén", "Saud Maia Leite"]
 years = [2026]
 
@@ -112,7 +115,7 @@ $$P_{n+1}(x) = x \sum_{k \leq n} A(n+1,k)\, P_k(x).$$
 When $A$ counts weighted chains in a poset, $P_n$ enumerates the chains by
 length.
 
-**Theorem (Brändén–Saud Leite).** If $A$ is lower unitriangular
+**Theorem (Brändén–Saud Leite).** [@BrandenLeite2024x] If $A$ is lower unitriangular
 and totally nonnegative, then every $P_n$ has nonnegative coefficients and
 only real zeros in $[-1, 0]$, and $P_n$ interlaces $P_{n+1}$.
 
@@ -120,7 +123,7 @@ The matrices covered are exactly the resolvable ones: a lower-unitriangular
 matrix admits a resolution by nonnegative weights and monic polynomials if and
 only if it is totally nonnegative. The same conclusion holds for totally
 nonnegative matrices with a positive constant diagonal. See also the
-[lower-unitriangular characterization on symmetricfunctions.com](https://www.symmetricfunctions.com/polyaFrequency.htm#tnnLowerUnitriangularMatrix).
+[lower-unitriangular characterization on symmetricfunctions.com](sf:tnnLowerUnitriangularMatrix).
 
 Applied to power-series kernels built from Pólya frequency sequences, the
 theorem gives PF polynomials whose consecutive rows interlace, for two row
@@ -147,10 +150,7 @@ under the nonnegative combinations that assemble the chain polynomials.
 
 ## References
 
-P. Brändén and L. Saud Maia Leite, [“Totally nonnegative matrices, chain
-enumeration and zeros of polynomials,”](https://arxiv.org/abs/2412.06595)
-*Advances in Mathematics* 487 (2026), 110760,
-[doi:10.1016/j.aim.2025.110760](https://doi.org/10.1016/j.aim.2025.110760).
+See [@BrandenLeite2024x].
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in
diff --git a/RealRooted/Challenges/BrandenSolus.lean b/RealRooted/Challenges/BrandenSolus.lean
index 4ffbd55e1..f4801cde4 100644
--- a/RealRooted/Challenges/BrandenSolus.lean
+++ b/RealRooted/Challenges/BrandenSolus.lean
@@ -12,6 +12,9 @@ version = 1
 section = "theorems"
 slug = "branden-solus"
 formalized = "2026-10-02"
+references = [
+  "BrandenSolus2019",
+]
 authors = ["Brändén", "Solus"]
 years = [2019]
 
@@ -63,7 +66,8 @@ $R_d(f) \ll f$ for the associated polynomial $f$ (`fPolynomial d p`).
 
 The derangement transform $D : x^j \mapsto d_j(x)$ sends every polynomial with nonnegative
 coordinates in the basis $x^k(1+x)^{d-k}$ to a real-rooted polynomial lying between the
-Eulerian polynomial $A_d$ and the derangement polynomial $d_d$ (Brändén–Solus, Corollary 3.7).
+Eulerian polynomial $A_d$ and the derangement polynomial $d_d$
+[@BrandenSolus2019, Cor. 3.7].
 Here it is derived from the Brändén–Vecchi Chow-resolution theorem for the Pascal matrix,
 whose resolving rows are exactly $x^k(1+x)^{d-k}$.
 
@@ -74,14 +78,12 @@ The proof establishes the excedance recurrence by removing the letter $0$ from a
 
 ## References
 
-P. Brändén and L. Solus, “Symmetric decompositions and real-rootedness,”
-*International Mathematics Research Notices* 2021 (2021), no. 10, 7764–7798,
-[doi:10.1093/imrn/rnz059](https://doi.org/10.1093/imrn/rnz059). See the
-[symmetric-decomposition overview on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedInterlacing.htm#symmetricIDecomposition).
+See [@BrandenSolus2019] and the
+[symmetric-decomposition overview on symmetricfunctions.com](sf:symmetricIDecomposition).
 <!-- /realrooted-catalog-content -->
 
 Human statement:
-https://www.symmetricfunctions.com/realRootedInterlacing.htm#symmetricIDecomposition
+sf:symmetricIDecomposition
 
 Original publication: P. Branden and L. Solus, "Symmetric decompositions and
 real-rootedness", International Mathematics Research Notices 2021 (2019),
diff --git a/RealRooted/Challenges/BrandenVecchi.lean b/RealRooted/Challenges/BrandenVecchi.lean
index 9bafaae12..5815dd1f3 100644
--- a/RealRooted/Challenges/BrandenVecchi.lean
+++ b/RealRooted/Challenges/BrandenVecchi.lean
@@ -12,6 +12,9 @@ version = 1
 section = "theorems"
 slug = "branden-vecchi"
 formalized = "2026-09-12"
+references = [
+  "BrandenVecchi2025x",
+]
 authors = ["Brändén", "Vecchi"]
 years = [2025]
 
@@ -115,9 +118,9 @@ $$c_n = \sum_{k \leq n} A(n,k)\, d_k.$$
 
 For the incidence data of a poset, these generalize the Chow polynomials of
 matroids and posets; see the
-[Chow-polynomial definition on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedGraphs.htm#chowPolynomialDefinition).
+[Chow-polynomial definition on symmetricfunctions.com](sf:chowPolynomialDefinition).
 
-**Theorem (Brändén–Vecchi).** If $A$ is lower unitriangular
+**Theorem (Brändén–Vecchi).** [@BrandenVecchi2025x] If $A$ is lower unitriangular
 and totally nonnegative, then $c_n$ and $d_n$ have nonnegative coefficients
 and only real zeros. Moreover $c_n$ interlaces $c_{n+1}$, and $c_n$
 interlaces $d_n$.
@@ -137,9 +140,7 @@ Further results:
 
 ## References
 
-P. Brändén and L. Vecchi, “Chow polynomials of totally nonnegative matrices
-and posets” (2025).
-The resolution machinery comes from the
+See [@BrandenVecchi2025x]. The resolution machinery comes from the
 [Brändén–Saud Leite page](/RealRooted/theorems/branden-leite/).
 <!-- /realrooted-catalog-content -->
 
diff --git a/RealRooted/Challenges/CauchyInterlacing.lean b/RealRooted/Challenges/CauchyInterlacing.lean
index 470b18afe..a61853f16 100644
--- a/RealRooted/Challenges/CauchyInterlacing.lean
+++ b/RealRooted/Challenges/CauchyInterlacing.lean
@@ -9,6 +9,10 @@ version = 1
 section = "theorems"
 slug = "cauchy-interlacing"
 formalized = "2026-07-10"
+references = [
+  "Godsil2017",
+  "Fisk2005x",
+]
 authors = ["Fisk"]
 years = [2005]
 
@@ -49,14 +53,12 @@ from the Courant–Fischer min-max principle.
 
 ## References
 
-C. D. Godsil, *Algebraic Combinatorics*, Routledge, 2017; and S. Fisk,
-“A very short proof of Cauchy’s interlace theorem for eigenvalues of Hermitian
-matrices,” *American Mathematical Monthly* 112 (2005), 118.  See also the
-[Cauchy interlacing entry on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedInterlacing.htm#cauchyInterlacingTheorem).
+See [@Godsil2017; @Fisk2005x] and the
+[Cauchy interlacing entry on symmetricfunctions.com](sf:cauchyInterlacingTheorem).
 <!-- /realrooted-catalog-content -->
 
 Human statement:
-https://www.symmetricfunctions.com/realRootedInterlacing.htm#cauchyInterlacingTheorem
+sf:cauchyInterlacingTheorem
 
 References used by the catalog:
 
diff --git a/RealRooted/Challenges/ChordalClawFreeAcyclicSinks.lean b/RealRooted/Challenges/ChordalClawFreeAcyclicSinks.lean
index 98314bf89..167971a38 100644
--- a/RealRooted/Challenges/ChordalClawFreeAcyclicSinks.lean
+++ b/RealRooted/Challenges/ChordalClawFreeAcyclicSinks.lean
@@ -47,7 +47,7 @@ the vertices.
 
 A **reverse perfect elimination order** is a linear order of the vertices in
 which the earlier neighbors of every vertex form a clique. A finite graph has
-such an order exactly when it is [chordal](https://www.symmetricfunctions.com/realRootedGraphs.htm#chordalGraph);
+such an order exactly when it is [chordal](sf:chordalGraph);
 here the order is part of the hypothesis.
 
 **Theorem.** If a finite claw-free graph $G$ has a reverse perfect elimination
diff --git a/RealRooted/Challenges/ChudnovskySeymour.lean b/RealRooted/Challenges/ChudnovskySeymour.lean
index a6054e0c3..284ddfa86 100644
--- a/RealRooted/Challenges/ChudnovskySeymour.lean
+++ b/RealRooted/Challenges/ChudnovskySeymour.lean
@@ -10,6 +10,9 @@ version = 1
 section = "theorems"
 slug = "chudnovsky-seymour"
 formalized = "2026-07-09"
+references = [
+  "ChudnovskySeymour2007",
+]
 authors = ["Chudnovsky", "Seymour"]
 years = [2007]
 
@@ -34,19 +37,16 @@ headline = true
 
 The independence polynomial counts independent vertex sets by cardinality.
 Chudnovsky and Seymour proved that it is real-rooted for every finite
-claw-free graph.
+claw-free graph [@ChudnovskySeymour2007].
 
 ## References
 
-M. Chudnovsky and P. Seymour, “The roots of the independence polynomial of a
-clawfree graph,” *Journal of Combinatorial Theory, Series B* 97 (2007),
-350–357, [doi:10.1016/j.jctb.2006.06.001](https://doi.org/10.1016/j.jctb.2006.06.001).
-See also the
-[graph-polynomial context on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedGraphs.htm#clawFreeGraph).
+See [@ChudnovskySeymour2007] and the
+[graph-polynomial context on symmetricfunctions.com](sf:clawFreeGraph).
 <!-- /realrooted-catalog-content -->
 
 Human statement:
-https://www.symmetricfunctions.com/realRootedGraphs.htm#clawFreeGraph
+sf:clawFreeGraph
 
 Original publication: M. Chudnovsky and P. Seymour, "The roots of the
 independence polynomial of a clawfree graph", J. Combin. Theory Ser. B 97
diff --git a/RealRooted/Challenges/ClawFreeOrientationSinks.lean b/RealRooted/Challenges/ClawFreeOrientationSinks.lean
index 046656848..9b907bdcd 100644
--- a/RealRooted/Challenges/ClawFreeOrientationSinks.lean
+++ b/RealRooted/Challenges/ClawFreeOrientationSinks.lean
@@ -55,7 +55,7 @@ $2^{|E(G)| - \sum_{v \in S} \deg v}$.
 The right-hand side of the identity is a weighted independence polynomial with
 positive weights, which is real-rooted for claw-free graphs by the weighted form
 of the [Chudnovsky–Seymour theorem](/RealRooted/theorems/chudnovsky-seymour/).
-See also [claw-free graphs on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedGraphs.htm#clawFreeGraph).
+See also [claw-free graphs on symmetricfunctions.com](sf:clawFreeGraph).
 Line graphs are one claw-free subfamily.
 
 For acyclic orientations, see the pages on
diff --git a/RealRooted/Challenges/CoefficientDominance.lean b/RealRooted/Challenges/CoefficientDominance.lean
index f9d4a4885..c478d960d 100644
--- a/RealRooted/Challenges/CoefficientDominance.lean
+++ b/RealRooted/Challenges/CoefficientDominance.lean
@@ -14,6 +14,9 @@ version = 1
 section = "theorems"
 slug = "coefficient-dominance"
 formalized = "2026-09-01"
+references = [
+  "Kozlov2022",
+]
 
 [[theorems]]
 name = "RealRooted.CoefficientDominance.eval_neg_ne_zero_of_dominant"
@@ -132,12 +135,12 @@ the explicit correction sums under positivity. Finally, if positive
 `x (j - 1)` and `x j` lie on opposite sides of reciprocal endpoints, then
 `log_gap_ge` converts a ratio at least `exp 2` between those endpoints into a
 logarithmic gap of at least two.
+
 ## References
 
 These results were extracted from a formal proof that the rows of
-[OEIS A366159](https://oeis.org/A366159) are real-rooted and interlace.  The
-triangle appears in D. N. Kozlov, [“Stirling
-complexes,”](https://arxiv.org/abs/2309.17142) arXiv:2309.17142 (2023).
+[OEIS A366159](https://oeis.org/A366159) are real-rooted and interlace. The triangle
+appears in [@Kozlov2022].
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/CommonInterleaver.lean b/RealRooted/Challenges/CommonInterleaver.lean
index 0fe629128..a31d91ecd 100644
--- a/RealRooted/Challenges/CommonInterleaver.lean
+++ b/RealRooted/Challenges/CommonInterleaver.lean
@@ -15,6 +15,11 @@ version = 1
 section = "concepts"
 slug = "common-interleavers"
 formalized = "2026-07-03"
+references = [
+  "ChudnovskySeymour2007",
+  "Branden2015",
+  "Leander2016x",
+]
 authors = ["Chudnovsky", "Seymour", "Leander"]
 years = [2007, 2016]
 
@@ -114,7 +119,8 @@ interleaver** if a single real-rooted $h$ satisfies $f_i \ll h$ for every $i$.
 The family is **compatible** if every nonnegative combination
 $c_1 f_1 + \dotsb + c_m f_m$ is zero or real-rooted.
 
-**Theorem (Chudnovsky–Seymour).** For real-rooted polynomials with positive
+**Theorem (Chudnovsky–Seymour).** [@ChudnovskySeymour2007] For real-rooted
+polynomials with positive
 leading coefficients, the following are equivalent:
 
 1. every pair is compatible;
@@ -136,7 +142,7 @@ This is the interlacing input for the real-rootedness of independence
 polynomials of claw-free graphs; see the
 [Chudnovsky–Seymour page](/RealRooted/theorems/chudnovsky-seymour/).
 
-**Theorem (Leander).** Let $f_0, \dotsc, f_{n-1}$ have positive leading
+**Theorem (Leander).** [@Leander2016x, Thm. 2.3] Let $f_0, \dotsc, f_{n-1}$ have positive leading
 coefficients, and suppose that $f_i$ and $f_j$ are compatible and $x f_i$ and
 $f_j$ are compatible whenever $i \le j$. Define
 $$
@@ -153,15 +159,7 @@ has largest root at most that of the polynomial at the root.
 
 ## References
 
-M. Chudnovsky and P. Seymour, “The roots of the independence polynomial of a
-clawfree graph,” *J. Combin. Theory Ser. B* 97 (2007), 350–357,
-[doi:10.1016/j.jctb.2006.06.001](https://doi.org/10.1016/j.jctb.2006.06.001).
-P. Brändén, [“Unimodality, log-concavity, real-rootedness and
-beyond,”](https://arxiv.org/abs/1410.6601) in *Handbook of Enumerative
-Combinatorics*, CRC Press (2015), Section 7.8.
-M. Leander, [“Compatible polynomials and edgewise
-subdivisions,”](https://arxiv.org/abs/1605.05287) arXiv:1605.05287 (2016),
-Theorem 2.3.
+See [@ChudnovskySeymour2007; @Branden2015, Sec. 7.8; @Leander2016x, Thm. 2.3].
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in
diff --git a/RealRooted/Challenges/CourantFischer.lean b/RealRooted/Challenges/CourantFischer.lean
index 044fc42d0..2780929aa 100644
--- a/RealRooted/Challenges/CourantFischer.lean
+++ b/RealRooted/Challenges/CourantFischer.lean
@@ -8,6 +8,10 @@ version = 1
 section = "theorems"
 slug = "courant-fischer"
 formalized = "2026-07-01"
+references = [
+  "Courant1920",
+  "Fischer1905",
+]
 authors = ["Fischer", "Courant"]
 years = [1905, 1920]
 
@@ -44,7 +48,7 @@ $$
 R_A(x) = \frac{\operatorname{Re}(x^* A x)}{\operatorname{Re}(x^* x)} .
 $$
 
-**Theorem** (Courant–Fischer). For every $k$ with $0 \leq k < N$:
+**Theorem** (Courant–Fischer). [@Courant1920; @Fischer1905] For every $k$ with $0 \leq k < N$:
 
 1. there is a subspace $W \subseteq \mathbb K^N$ of dimension $k + 1$ such that
    $R_A(x) \geq \lambda_k$ for every nonzero $x \in W$;
@@ -75,11 +79,7 @@ theorem](/RealRooted/theorems/cauchy-interlacing/) and Weyl's rank-one interlaci
 
 ## References
 
-R. Courant, “Über die Eigenwerte bei den Differentialgleichungen der mathematischen
-Physik,” *Math. Z.* 7 (1920), 1–57,
-[doi:10.1007/bf01199396](https://doi.org/10.1007/bf01199396); E. Fischer, “Über quadratische
-Formen mit reellen Koeffizienten,” *Monatsh. Math. Phys.* 16 (1905), 234–249,
-[doi:10.1007/bf01693781](https://doi.org/10.1007/bf01693781).
+See [@Courant1920; @Fischer1905].
 <!-- /realrooted-catalog-content -->
 
 This module exposes the Courant–Fischer principle for Hermitian matrices and its
diff --git a/RealRooted/Challenges/DecoEulerian.lean b/RealRooted/Challenges/DecoEulerian.lean
index a7e61f344..c7f30c3ea 100644
--- a/RealRooted/Challenges/DecoEulerian.lean
+++ b/RealRooted/Challenges/DecoEulerian.lean
@@ -11,6 +11,9 @@ version = 1
 section = "theorems"
 slug = "deco-eulerian-transform"
 formalized = "2026-10-05"
+references = [
+  "BarcucciLungoPinzani1996",
+]
 authors = ["Alexandersson"]
 years = [2026]
 
@@ -86,12 +89,8 @@ input over $\mathbb R$ completes the proof.
 
 ## References
 
-P. Alexandersson, *An Eulerian deformation from deco polyominoes*, manuscript
-(2026).
-
-E. Barcucci, A. Del Lungo, and R. Pinzani, [“Deco polyominoes, permutations
-and random generation,”](https://doi.org/10.1016/0304-3975(95)00199-9)
-*Theoretical Computer Science* 159 (1996), 29–42.
+P. Alexandersson, *An Eulerian deformation from deco polyominoes*, manuscript (2026).
+For the deco-polyomino background, see [@BarcucciLungoPinzani1996].
 <!-- /realrooted-catalog-content -->
 
 This module exposes the checked interval-preservation theorem. The energy,
diff --git a/RealRooted/Challenges/DerivativeInterlacing.lean b/RealRooted/Challenges/DerivativeInterlacing.lean
index adc8c39f0..87ad4a1ca 100644
--- a/RealRooted/Challenges/DerivativeInterlacing.lean
+++ b/RealRooted/Challenges/DerivativeInterlacing.lean
@@ -13,6 +13,11 @@ version = 1
 section = "theorems"
 slug = "derivative-interlacing"
 formalized = "2026-05-12"
+references = [
+  "Fisk2006x",
+  "Obreschkoff1963",
+  "Jensen1913",
+]
 authors = ["Rolle", "Laguerre", "Jensen", "Obreschkoff", "Fisk"]
 years = [1691, 1882, 1913, 1963, 2006]
 
@@ -71,7 +76,7 @@ $f' \ll g'$. Here we use the zero-aware relation `Interl`, which holds
 whenever one side vanishes; if $f$ is nonconstant, both derivatives are
 nonzero and $f' \ll g'$ holds in the strict sense `StrictInterl`.
 
-**Theorem** (Jensen). Let $p$ be a nonconstant real polynomial. Every
+**Theorem** (Jensen). [@Jensen1913] Let $p$ be a nonconstant real polynomial. Every
 non-real zero of $p'$ lies in a *Jensen disk* of $p$: the closed disk whose
 diameter is the segment from a non-real zero $z$ of $p$ to $\bar z$.  In
 particular, if $p$ is real-rooted then so is $p'$.
@@ -89,7 +94,8 @@ further zero of $f'$ strictly between any two consecutive distinct zeros of
 $f$. This gives $d - 1$ zeros of $f'$, counted with multiplicity, in the
 required positions, and $f'$ has no others since its degree is $d - 1$.
 
-For the second theorem, we use Obreschkoff's theorem: $f \ll g$ implies that
+For the second theorem, we use Obreschkoff's theorem [@Obreschkoff1963]:
+$f \ll g$ implies that
 every real linear combination $\lambda f + \mu g$ is zero or real-rooted.
 Differentiating the combinations shows that the same holds for $f'$ and
 $g'$, and the converse direction of Obreschkoff's theorem returns
@@ -102,14 +108,9 @@ imaginary part of sign opposite to $\operatorname{Im} w$, which is impossible.
 
 ## References
 
-S. Fisk, [*Polynomials, roots, and
-interlacing*](https://arxiv.org/abs/math/0612833), arXiv:math/0612833 (2006),
-Chapter 1; N. Obreschkoff, *Verteilung und Berechnung der Nullstellen reeller
-Polynome*, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963. J. L. W. V. Jensen,
-*Recherches sur la théorie des équations*, *Acta Mathematica* 36 (1913), 181–195,
-[doi:10.1007/BF02422380](https://doi.org/10.1007/BF02422380). The
-statement $f' \ll f$ is the remark after
-[Wagner's lemma on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedInterlacing.htm#wagnerLemma).
+For derivative interlacing, see [@Fisk2006x, Ch. 1] and [@Obreschkoff1963].
+Jensen's theorem is [@Jensen1913]. The statement $f' \ll f$ is the remark after
+[Wagner's lemma on symmetricfunctions.com](sf:wagnerLemma).
 <!-- /realrooted-catalog-content -->
 
 This module exposes Rolle's theorem, derivative interlacing in all positive
diff --git a/RealRooted/Challenges/Descartes.lean b/RealRooted/Challenges/Descartes.lean
index 48d340d20..1cbe33453 100644
--- a/RealRooted/Challenges/Descartes.lean
+++ b/RealRooted/Challenges/Descartes.lean
@@ -62,7 +62,7 @@ résolution des équations numériques*, 1807; J. Fourier, *Analyse des
 équations déterminées*, 1831. See also the
 [contextual account on symmetricfunctions.com][descartes-context].
 
-[descartes-context]: https://www.symmetricfunctions.com/realRooted.htm#descartesRuleOfSigns
+[descartes-context]: sf:descartesRuleOfSigns
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in
diff --git a/RealRooted/Challenges/EdreiThoma.lean b/RealRooted/Challenges/EdreiThoma.lean
index 0777fc672..d1f58732e 100644
--- a/RealRooted/Challenges/EdreiThoma.lean
+++ b/RealRooted/Challenges/EdreiThoma.lean
@@ -9,6 +9,10 @@ version = 1
 section = "theorems"
 slug = "edrei-thoma"
 formalized = "2026-10-13"
+references = [
+  "AissenSchoenbergWhitney1952",
+  "Edrei1953",
+]
 
 [[theorems]]
 name = "RealRooted.EdreiThoma.exists_canonicalProduct_representation"
@@ -127,20 +131,16 @@ Pólya-frequency sequence with $a_0 = 1$ arises this way: there are
 $\gamma \ge 0$ and nonnegative $\alpha_i, \beta_i$ with
 $\sum_i(\alpha_i+\beta_i) < \infty$ such that the generating function equals
 $e^{\gamma z}\prod_i(1+\alpha_i z)/\prod_i(1-\beta_i z)$ near $0$. The proof
-follows Edrei's determinant identities and monotone limits, but replaces his
+follows Edrei's determinant identities [@Edrei1953] and monotone limits, but replaces his
 use of Nevanlinna's second main theorem by an elementary growth argument
 (Jensen's formula and a Borel-type lemma). Together the two directions give the
 classification of all one-sided Pólya-frequency sequences.
 
 ## References
 
-M. Aissen, I. J. Schoenberg, and A. M. Whitney, “On the generating functions
-of totally positive sequences. I,” *Journal d’Analyse Mathématique* 2 (1952),
-93–103, [doi:10.1007/BF02786970](https://doi.org/10.1007/BF02786970), prove the easy direction.
-The converse is due to A. Edrei (1952); we
-follow his proof in *Canadian Journal of Mathematics* 5 (1953), 86–94,
-[doi:10.4153/CJM-1953-010-3](https://doi.org/10.4153/CJM-1953-010-3). See the
-[Edrei–Thoma theorem on symmetricfunctions.com](https://www.symmetricfunctions.com/polyaFrequency.htm#edreiThomaTheorem).
+The easy direction is [@AissenSchoenbergWhitney1952]. The converse is due to Edrei;
+we follow [@Edrei1953]. See the
+[Edrei–Thoma theorem on symmetricfunctions.com](sf:edreiThomaTheorem).
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/EndpointDerivative.lean b/RealRooted/Challenges/EndpointDerivative.lean
index 31bd6cce9..0ad8ecb7a 100644
--- a/RealRooted/Challenges/EndpointDerivative.lean
+++ b/RealRooted/Challenges/EndpointDerivative.lean
@@ -9,6 +9,9 @@ version = 1
 section = "theorems"
 slug = "endpoint-derivative"
 formalized = "2026-09-26"
+references = [
+  "Fisk2006x",
+]
 
 [[theorems]]
 name = "RealRooted.strictInterl_endpointDerivative"
@@ -112,12 +115,10 @@ for both operators.
 
 ## References
 
-S. Fisk, [*Polynomials, roots, and interlacing*](https://arxiv.org/abs/math/0612833),
-arXiv:math/0612833 (2006), Chapter 1.
-
-The first applications are the rows of [OEIS A142071](https://oeis.org/A142071),
-[OEIS A172106](https://oeis.org/A172106), [OEIS A172107](https://oeis.org/A172107), and
-[OEIS A172108](https://oeis.org/A172108).
+For derivative interlacing, see [@Fisk2006x, Ch. 1]. The first applications
+are the rows of [OEIS A142071](https://oeis.org/A142071),
+[OEIS A172106](https://oeis.org/A172106), [OEIS A172107](https://oeis.org/A172107),
+and [OEIS A172108](https://oeis.org/A172108).
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/EulerBidiagonal.lean b/RealRooted/Challenges/EulerBidiagonal.lean
index 909aeb106..6ce028025 100644
--- a/RealRooted/Challenges/EulerBidiagonal.lean
+++ b/RealRooted/Challenges/EulerBidiagonal.lean
@@ -15,6 +15,11 @@ version = 1
 section = "theorems"
 slug = "euler-bidiagonal-interlacing"
 formalized = "2026-10-09"
+references = [
+  "AndrewsGawronskiLittlejohn2011",
+  "AndrewsEggeGawronskiLittlejohn2013",
+  "Mongelli2012",
+]
 years = [2026]
 
 [[definitions]]
@@ -125,8 +130,9 @@ negative roots, with $\deg g = \deg f + 1$ and no common root. Then $Tf \ll Tg$,
 without common roots. In particular consecutive rows strictly interlace:
 $P_n \ll P_{n+1}$ for all $n$.
 
-Real-rootedness of these rows is classical (Andrews–Gawronski–Littlejohn;
-Andrews–Egge–Gawronski–Littlejohn; Mongelli). Interlacing of consecutive rows, and the
+Real-rootedness of these rows is classical
+[@AndrewsGawronskiLittlejohn2011; @AndrewsEggeGawronskiLittlejohn2013;
+@Mongelli2012]. Interlacing of consecutive rows, and the
 preservation of interlacing pairs, appear to be new. They were found and formalized in this
 project; see issue #1324. The usual first-order step theorems (Liu–Wang, Wang–Yeh) do not
 apply, because the step is second order in $\theta$. Nor does $T$ preserve real-rootedness on
@@ -165,16 +171,8 @@ real-rooted, and the Obreschkoff converse gives $Tf \ll Tg$.
 
 ## References
 
-G. E. Andrews, W. Gawronski and L. L. Littlejohn, “The Legendre–Stirling numbers,”
-*Discrete Mathematics* 311 (2011), 1255–1272,
-[doi:10.1016/j.disc.2011.02.028](https://doi.org/10.1016/j.disc.2011.02.028); G. E. Andrews, E. S. Egge, W. Gawronski and
-L. L. Littlejohn, “The Jacobi–Stirling numbers,” *Journal of Combinatorial Theory, Series A*
-120 (2013), 288–303, [doi:10.1016/j.jcta.2012.08.006](https://doi.org/10.1016/j.jcta.2012.08.006);
-P. Mongelli, “Total positivity properties of Jacobi–Stirling numbers,”
-*Advances in Applied Mathematics* 48 (2012), 354–364,
-[doi:10.1016/j.aam.2011.06.008](https://doi.org/10.1016/j.aam.2011.06.008).
-
-The Jacobi–Stirling triangles are on the
+See [@AndrewsGawronskiLittlejohn2011; @AndrewsEggeGawronskiLittlejohn2013;
+@Mongelli2012]. The Jacobi–Stirling triangles are on the
 [Jacobi–Stirling page](/RealRooted/families/jacobi-stirling-numbers/).
 <!-- /realrooted-catalog-content -->
 
diff --git a/RealRooted/Challenges/Eulerian.lean b/RealRooted/Challenges/Eulerian.lean
index 529243d14..7af357207 100644
--- a/RealRooted/Challenges/Eulerian.lean
+++ b/RealRooted/Challenges/Eulerian.lean
@@ -14,6 +14,9 @@ version = 1
 section = "families"
 slug = "eulerian"
 formalized = "2026-07-10"
+references = [
+  "Brenti1994",
+]
 authors = ["Frobenius", "Brenti"]
 years = [1910, 1994]
 
@@ -115,23 +118,21 @@ itself is still not formalized.
 
 ## References
 
-The ordinary Eulerian recurrence and its real-rootedness go back to
-F. G. Frobenius, “Über die Bernoullischen Zahlen und die Eulerschen Polynome,”
-Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften
-(1910), 809–847.  For type $B$, see F. Brenti, [“q-Eulerian polynomials arising
-from Coxeter groups,”](https://doi.org/10.1006/eujc.1994.1046) *European Journal
-of Combinatorics* 15 (1994), 417–441.  See also the
-[Eulerian polynomials on symmetricfunctions.com](https://www.symmetricfunctions.com/eulerian.htm#eulerianPolynomial).
+The ordinary Eulerian recurrence and its real-rootedness go back to F. G. Frobenius,
+“Über die Bernoullischen Zahlen und die Eulerschen Polynome,” *Sitzungsberichte der
+Königlich Preussischen Akademie der Wissenschaften* (1910), 809–847. For type $B$,
+see [@Brenti1994]. See also the
+[Eulerian polynomials on symmetricfunctions.com](sf:eulerianPolynomial).
 <!-- /realrooted-catalog-content -->
 
 Human statements:
 
 * Eulerian polynomials:
-  https://www.symmetricfunctions.com/eulerian.htm#eulerianPolynomial
+  sf:eulerianPolynomial
 * Eulerian Sturm sequence example:
-  https://www.symmetricfunctions.com/realRootedWords.htm#ex:eulerianSturm
+  sf:ex:eulerianSturm
 * Type `B` Eulerian polynomials:
-  https://www.symmetricfunctions.com/realRootedWords.htm#typeBEulerianPolynomial
+  sf:typeBEulerianPolynomial
 
 Classical origin: F. G. Frobenius, "Über die Bernoullischen Zahlen und die
 Eulerschen Polynome", Sitzungsberichte der Königlich Preussischen Akademie der
diff --git a/RealRooted/Challenges/EulerianVariations.lean b/RealRooted/Challenges/EulerianVariations.lean
index 12f5cd350..282d3d3aa 100644
--- a/RealRooted/Challenges/EulerianVariations.lean
+++ b/RealRooted/Challenges/EulerianVariations.lean
@@ -11,6 +11,9 @@ version = 1
 section = "theorems"
 slug = "eulerian-variations"
 formalized = "2026-09-09"
+references = [
+  "Alexandersson2026Eulerian",
+]
 authors = ["Alexandersson", "Shankar"]
 years = [2026]
 
@@ -105,8 +108,7 @@ statements are about the polynomials themselves.
 
 ## References
 
-P. Alexandersson, [“Real-rooted Eulerian polynomials from permutations,
-words, and paths,”](https://arxiv.org/abs/2609.07325) arXiv:2609.07325 (2026).
+See [@Alexandersson2026Eulerian].
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in
diff --git a/RealRooted/Challenges/FPolynomialRecurrences.lean b/RealRooted/Challenges/FPolynomialRecurrences.lean
index d28214a32..8ccf802be 100644
--- a/RealRooted/Challenges/FPolynomialRecurrences.lean
+++ b/RealRooted/Challenges/FPolynomialRecurrences.lean
@@ -9,6 +9,9 @@ version = 1
 section = "theorems"
 slug = "f-transform-recurrences"
 formalized = "2026-10-11"
+references = [
+  "BrandenSolus2019",
+]
 
 [[theorems]]
 name = "RealRooted.fPolynomial_derivative"
@@ -148,16 +151,11 @@ theorem.
 
 ## References
 
-P. Brändén and L. Solus, “Symmetric decompositions and real-rootedness,”
-*International Mathematics Research Notices* 2021 (2021), 7764–7798,
-[doi:10.1093/imrn/rnz059](https://doi.org/10.1093/imrn/rnz059). See the
-[symmetric-decomposition overview on symmetricfunctions.com][bs-overview].
-
+See [@BrandenSolus2019] and the
+[symmetric-decomposition overview on symmetricfunctions.com](sf:symmetricIDecomposition).
 For the alternating-runs families and their references, see OEIS
 [A008970](https://oeis.org/A008970), [A059427](https://oeis.org/A059427), and
 [A186370](https://oeis.org/A186370).
-
-[bs-overview]: https://www.symmetricfunctions.com/realRootedInterlacing.htm#symmetricIDecomposition
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/FactorialCompression.lean b/RealRooted/Challenges/FactorialCompression.lean
index 29ef9019c..a02c4eeb0 100644
--- a/RealRooted/Challenges/FactorialCompression.lean
+++ b/RealRooted/Challenges/FactorialCompression.lean
@@ -9,6 +9,9 @@ version = 1
 section = "theorems"
 slug = "factorial-compression"
 formalized = "2026-10-09"
+references = [
+  "Zhang2026SeparableInterlacing",
+]
 authors = ["Zhanhe Zhang"]
 years = [2026]
 
@@ -76,12 +79,11 @@ transfers strict interlacing through the composition by the strict Schur–Szeg
 endpoint: composition with a multiplier of the right degree having only negative
 roots preserves strict interlacing of polynomials without common roots.  The
 adjacent-degree lifting to gamma transforms is a separate step, used for the
-separable permutations.
+separable permutations in Zhang's argument [@Zhang2026SeparableInterlacing].
 
 ## References
 
-Z. Zhang, [“Strict interlacing of the descent polynomials of separable
-permutations,”](https://doi.org/10.2139/ssrn.7510941) SSRN preprint (2026). The Lean development follows the draft
+See [@Zhang2026SeparableInterlacing]. The Lean development follows the draft
 formalization of PR #1132 by yyou59548-design.
 <!-- /realrooted-catalog-content -->
 
diff --git a/RealRooted/Challenges/Favard.lean b/RealRooted/Challenges/Favard.lean
index 3c8851a60..10ad10a27 100644
--- a/RealRooted/Challenges/Favard.lean
+++ b/RealRooted/Challenges/Favard.lean
@@ -9,6 +9,9 @@ version = 1
 section = "theorems"
 slug = "favard"
 formalized = "2026-09-07"
+references = [
+  "Favard1935",
+]
 authors = ["Favard"]
 years = [1935]
 
@@ -62,7 +65,7 @@ $$
 If every $\beta_{n+1}$ is positive, the real polynomials $P_n$ are nonzero and
 real-rooted, and consecutive polynomials interlace.
 
-**Favard’s theorem.** Over an integral domain $R$, the monic polynomials $P_n$
+**Favard’s theorem.** [@Favard1935] Over an integral domain $R$, the monic polynomials $P_n$
 form a basis of $R[x]$. Let $L$ be the linear functional that extracts the
 coefficient of $P_0$ in this basis, and let $\langle p, q \rangle = L(pq)$.
 
@@ -74,13 +77,12 @@ coefficient of $P_0$ in this basis, and let $\langle p, q \rangle = L(pq)$.
 
 ## References
 
-J. Favard, “Sur les polynômes de Tchebicheff,” *Comptes rendus de l’Académie
-des sciences* 200 (1935), 2052–2053.  See also the
-[interlacing overview on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedInterlacing.htm#favardInterlacing).
+See [@Favard1935] and the
+[interlacing overview on symmetricfunctions.com](sf:favardInterlacing).
 <!-- /realrooted-catalog-content -->
 
 Human statement:
-https://www.symmetricfunctions.com/realRootedInterlacing.htm#favardInterlacing
+sf:favardInterlacing
 
 Original publication: J. Favard, "Sur les polynomes de Tchebicheff",
 C. R. Acad. Sci. Paris 200 (1935), 2052--2053.
diff --git a/RealRooted/Challenges/FiniteFreeConvolutions.lean b/RealRooted/Challenges/FiniteFreeConvolutions.lean
index 082cf5efd..32a882bed 100644
--- a/RealRooted/Challenges/FiniteFreeConvolutions.lean
+++ b/RealRooted/Challenges/FiniteFreeConvolutions.lean
@@ -13,6 +13,10 @@ version = 1
 section = "theorems"
 slug = "finite-free-convolutions"
 formalized = "2026-09-11"
+references = [
+  "MarcusSpielmanSrivastava2022",
+  "GribinskiMarcus2022",
+]
 authors = ["Marcus", "Spielman", "Srivastava", "Gribinski"]
 years = [2022]
 
@@ -108,10 +112,10 @@ lists $p_k$, $q_k$.
 **Theorem.**
 
 - If $p$ and $q$ are real-rooted of degree at most $d$, then $p \boxplus_d q$ is
-  real-rooted (Marcus, Spielman, Srivastava).
+  real-rooted [@MarcusSpielmanSrivastava2022].
 - If $f$ and $g$ have degree exactly $n$, positive leading coefficients, and
   only nonnegative real roots, then $f \boxplus_{m,n} g$ is zero or has only
-  nonnegative real roots (Gribinski, Marcus).
+  nonnegative real roots [@GribinskiMarcus2022].
 - The same holds for the parameters $\alpha = -\tfrac12$ and
   $\alpha = \tfrac12$, for inputs of degree at most $n$ that are real-rooted with
   nonnegative roots.
@@ -134,16 +138,10 @@ composition theorem on the Hadamard page applies.
 
 ## References
 
-A. W. Marcus, D. A. Spielman, and N. Srivastava, “Finite free convolutions of
-polynomials,” *Probability Theory and Related Fields* 182 (2022), 807–848,
-[doi:10.1007/s00440-021-01105-w](https://doi.org/10.1007/s00440-021-01105-w);
-A. Gribinski and A. W. Marcus, [“A rectangular additive convolution for
-polynomials,”](https://doi.org/10.5070/c62156888) *Combinatorial Theory* 2(1)
-(2022), #16.
-For the application to the Narayana transformation, see
-[symmetricfunctions.com][mao-wang].
+See [@MarcusSpielmanSrivastava2022; @GribinskiMarcus2022]. For the application to
+the Narayana transformation, see [symmetricfunctions.com][mao-wang].
 
-[mao-wang]: https://www.symmetricfunctions.com/realRooted.htm#thm:narayanaTransformationMaoWang
+[mao-wang]: sf:thm:narayanaTransformationMaoWang
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade. The proofs live in
diff --git a/RealRooted/Challenges/FiniteMemoryDefect.lean b/RealRooted/Challenges/FiniteMemoryDefect.lean
index 412d58e91..3950305d9 100644
--- a/RealRooted/Challenges/FiniteMemoryDefect.lean
+++ b/RealRooted/Challenges/FiniteMemoryDefect.lean
@@ -8,6 +8,10 @@ version = 1
 section = "theorems"
 slug = "finite-memory-defect-bridges"
 formalized = "2026-10-11"
+references = [
+  "BogvadHagg2017",
+  "NyandwiHaggKurujyibwamiUwimbabazi2026x",
+]
 
 [[theorems]]
 name = "RealRooted.HasNonnegCoeffs.natDegree_add_eq_max"
@@ -81,14 +85,7 @@ $3d+K_{n\bmod3}=2n$.
 
 ## References
 
-R. Bøgvad and C. Hägg, “A refinement for rational functions of Pólya's method
-to construct Voronoi diagrams,” *Journal of Mathematical Analysis and
-Applications* 452(1) (2017), 312–334,
-[doi:10.1016/j.jmaa.2017.02.071](https://doi.org/10.1016/j.jmaa.2017.02.071);
-B. Nyandwi, C. Hägg, C. Kurujyibwami, and L. F. R. Uwimbabazi, “Voronoi limit
-measures for iterates of constant-coefficient differential operators on
-rational functions with simple poles,”
-[arXiv:2604.05189](https://arxiv.org/abs/2604.05189).
+See [@BogvadHagg2017; @NyandwiHaggKurujyibwamiUwimbabazi2026x].
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/GammaTransform.lean b/RealRooted/Challenges/GammaTransform.lean
index 9aa1c904a..ad7b49b16 100644
--- a/RealRooted/Challenges/GammaTransform.lean
+++ b/RealRooted/Challenges/GammaTransform.lean
@@ -10,6 +10,11 @@ version = 1
 section = "theorems"
 slug = "gamma-transform"
 formalized = "2026-10-06"
+references = [
+  "Petersen2015EulerianNumbers",
+  "HosterStump2025x",
+  "Branden2015",
+]
 authors = ["Petersen", "Hoster", "Stump"]
 years = [2015, 2025]
 
@@ -80,7 +85,8 @@ nonzero and real-rooted with only nonpositive zeros. In particular, if
 $\gamma \neq 0$ is real-rooted with nonnegative coefficients, then
 $\Gamma_d(\gamma)$ is real-rooted.
 
-**Theorem** (Hoster–Stump). Let $\deg \gamma \leq d/2$ and
+**Theorem** (Hoster–Stump). [@HosterStump2025x, Prop. 2.5] Let
+$\deg \gamma \leq d/2$ and
 $\deg \delta \leq (d+1)/2$, and let $\gamma$ and $\delta$ have nonnegative
 coefficients and nonzero constant terms. Then
 $\Gamma_d(\gamma) \ll \Gamma_{d+1}(\delta)$ if and only if
@@ -115,7 +121,8 @@ at a time. The Hoster–Stump equivalence follows by comparing the sorted root
 lists through the same monotone root map.
 
 For the pencil, the operator $T_n$ has a stable algebraic symbol, so the
-finite Borcea–Brändén theorem shows that it preserves real-rootedness on its
+finite Borcea–Brändén theorem [@Branden2015] shows that it preserves
+real-rootedness on its
 degree box. It therefore maps the real pencil spanned by $U_n$ and $V_n$ into
 the next one, starting from the linear pencil $a + bx$. By Obreschkoff's
 theorem, the pencil property makes $U_n$ and $V_n$ interlace, and the
@@ -123,12 +130,9 @@ constant terms $U_n(0) = 1$ and $V_n(0) = 0$ fix the orientation.
 
 ## References
 
-T. K. Petersen, *Eulerian Numbers*, Birkhäuser, 2015, Section 4.6;
-Hoster and Stump, [arXiv:2508.15538](https://arxiv.org/abs/2508.15538) (2025),
-Proposition 2.5; P. Brändén, “Unimodality, log-concavity,
-real-rootedness and beyond,” *Handbook of Enumerative Combinatorics*, CRC
-Press, 2015. See the
-[gamma polynomial on symmetricfunctions.com](https://www.symmetricfunctions.com/gammaPositivity.htm#gammaPolynomial).
+See [@Petersen2015EulerianNumbers, Sec. 4.6],
+[@HosterStump2025x, Prop. 2.5], and [@Branden2015]. See also the
+[gamma polynomial on symmetricfunctions.com](sf:gammaPolynomial).
 <!-- /realrooted-catalog-content -->
 
 This module exposes the real-rootedness transfer for the gamma transform, the
diff --git a/RealRooted/Challenges/GarloffWagnerHurwitz.lean b/RealRooted/Challenges/GarloffWagnerHurwitz.lean
index a9d6a2ae5..56f6632e7 100644
--- a/RealRooted/Challenges/GarloffWagnerHurwitz.lean
+++ b/RealRooted/Challenges/GarloffWagnerHurwitz.lean
@@ -8,6 +8,9 @@ version = 1
 section = "theorems"
 slug = "garloff-wagner-hurwitz"
 formalized = "2026-06-10"
+references = [
+  "GarloffWagner1996",
+]
 authors = ["Garloff", "Wagner"]
 years = [1996]
 
@@ -35,24 +38,23 @@ A real polynomial is *Hurwitz stable* if its coefficients are nonnegative and it
 zero with positive real part. The Hadamard product of $a = \sum_k a_k x^k$ and
 $b = \sum_k b_k x^k$ is $a \ast b = \sum_k a_k b_k x^k$.
 
-**Theorem (Garloff–Wagner).** If $a$ and $b$ are Hurwitz stable and $a \ast b \neq 0$,
+**Theorem (Garloff–Wagner).** [@GarloffWagner1996, Thm. 1] If $a$ and $b$ are
+Hurwitz stable and $a \ast b \neq 0$,
 then $a \ast b$ is Hurwitz stable.
 
 The proof writes $a = q(x^2) + x\,p(x^2)$. By the Hermite–Biehler theorem, $a$ is Hurwitz
 stable exactly when the odd and even parts interlace, $p \ll q$, or one of them vanishes
 and the other is a PF polynomial (real-rooted with nonnegative coefficients). The
 Hadamard product acts on the two parts separately, and both alternatives are preserved:
-interlacing by Garloff and Wagner's theorem on Hadamard products of interlacing pairs, and
+interlacing by Garloff and Wagner's theorem on Hadamard products of interlacing pairs
+[@GarloffWagner1996], and
 PF polynomials by Maló's theorem. The coefficient condition $a \ast b \neq 0$ excludes
 products of polynomials with disjoint supports, which vanish.
 
 ## References
 
-J. Garloff and D. G. Wagner, “Hadamard products of stable polynomials are stable,”
-*Journal of Mathematical Analysis and Applications* 202 (1996), 797–809,
-[doi:10.1006/jmaa.1996.0348](https://doi.org/10.1006/jmaa.1996.0348), Theorem 1. See
-the [Hadamard-product overview](https://www.symmetricfunctions.com/realRooted.htm#hadamardProductTheorems)
-on symmetricfunctions.com.
+See [@GarloffWagner1996, Thm. 1] and the
+[Hadamard-product overview](sf:hadamardProductTheorems) on symmetricfunctions.com.
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade. The proof lives in
diff --git a/RealRooted/Challenges/GeneralizedSnakePosets.lean b/RealRooted/Challenges/GeneralizedSnakePosets.lean
index 9c7ada042..c76291e4b 100644
--- a/RealRooted/Challenges/GeneralizedSnakePosets.lean
+++ b/RealRooted/Challenges/GeneralizedSnakePosets.lean
@@ -14,6 +14,10 @@ version = 1
 section = "families"
 slug = "generalized-snake-posets"
 formalized = "2026-10-05"
+references = [
+  "BraunJal2026x",
+  "AlexanderssonJal2024x",
+]
 authors = ["Braun", "Jal", "Alexandersson"]
 years = [2024, 2026]
 
@@ -111,13 +115,16 @@ A generalized snake poset $P(w)$ is a width-two poset built from a word $w$ in
 the letters $L$ and $R$. Braun and Jal show that the $h^*$-polynomial of its
 order polytope is the non-nesting rook polynomial $M_w$ of a skew board whose
 cells are the incomparable cross-chain pairs of $P(w)$. The board comes from
-the Alexandersson–Jal correspondence between naturally labeled width-two posets
+the Alexandersson–Jal correspondence [@AlexanderssonJal2024x] between
+naturally labeled width-two posets
 and skew shapes, under which the $P$-Eulerian polynomial is the non-nesting rook
 polynomial. The identification with the $h^*$-polynomial is
-cited from Braun–Jal and is not formalized. Lean formalizes the rook model:
+cited from Braun–Jal [@BraunJal2026x] and is not formalized. Lean formalizes
+the rook model:
 the board, its non-nesting rook polynomial $M_w$, and the recurrence below.
 
-**Theorem (Braun–Jal).** If $w$ is nonempty, then $M_w$ is real-rooted, and
+**Theorem (Braun–Jal).** [@BraunJal2026x] If $w$ is nonempty, then $M_w$ is
+real-rooted, and
 deleting the last letter of $w$ gives a polynomial interlacing $M_w$.
 
 The proof uses the **modified Narayana polynomials** $P_n = t^{-1} N_{n+1}$,
@@ -140,14 +147,8 @@ Induction on the length of $w$ along the recurrence then gives the theorem.
 
 ## References
 
-Braun and Jal, [“Order polytopes of generalized snake posets are
-h*-real-rooted,”](https://arxiv.org/abs/2607.00922) arXiv:2607.00922 (2026);
-P. Alexandersson and A. Jal, [“Rook matroids and log-concavity of
-$P$-Eulerian polynomials,”](https://arxiv.org/abs/2410.00127) arXiv:2410.00127
-(2024), extended abstract in *Séminaire Lotharingien de Combinatoire* 93B
-(2025), Art. 59. See the
-[rook matroid section](https://www.symmetricfunctions.com/lattice-path-matroids.htm#rookMatroidPolynomials)
-of the lattice path matroids page on symmetricfunctions.com.
+See [@BraunJal2026x] and [@AlexanderssonJal2024x, Art. 59]. See also the
+[rook matroid section](sf:rookMatroidPolynomials) on symmetricfunctions.com.
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in
diff --git a/RealRooted/Challenges/GraceApolarity.lean b/RealRooted/Challenges/GraceApolarity.lean
index 24905bef9..c6c4bab1c 100644
--- a/RealRooted/Challenges/GraceApolarity.lean
+++ b/RealRooted/Challenges/GraceApolarity.lean
@@ -13,6 +13,15 @@ version = 1
 section = "theorems"
 slug = "grace-apolarity"
 formalized = "2026-10-06"
+references = [
+  "Grace1902",
+  "Walsh1922",
+  "Szego1922",
+  "RahmanSchmeisser2002",
+  "BorceaBranden2009II",
+  "LiebSokal1981",
+  "GribinskiMarcus2022",
+]
 authors = ["Grace", "Walsh", "Szegő", "Lieb", "Sokal", "Borcea", "Brändén"]
 years = [1902, 1922, 1981, 2009]
 
@@ -132,7 +141,7 @@ $$
 \sum_{k=0}^n (-1)^k \binom{n}{k} a_k b_{n-k} = 0 .
 $$
 
-**Theorem** (Grace). Let $F$ and $G$ be apolar polynomials of degree exactly
+**Theorem** (Grace). [@Grace1902] Let $F$ and $G$ be apolar polynomials of degree exactly
 $n$. If all zeros of $F$ lie in a closed disk $\{z : |z - c| \leq r\}$ with
 $r \geq 0$, or in a closed half-plane $\{z : \operatorname{Im} z \geq b\}$,
 then $G$ has a zero in the same set.
@@ -150,7 +159,8 @@ specialization $z_1 = \dotsb = z_n = z$ is $p(z)$.
 and no zeros in the open upper half-plane. Then $\operatorname{Pol}_n(p)$ is
 stable: it does not vanish when all $z_i$ lie in the open upper half-plane.
 
-**Theorem** (Grace–Walsh–Szegő). Let $P \in \mathbb{C}[z_1, \dotsc, z_n]$ be
+**Theorem** (Grace–Walsh–Szegő). [@Walsh1922; @Szego1922] Let
+$P \in \mathbb{C}[z_1, \dotsc, z_n]$ be
 symmetric and multiaffine. Then $P = \operatorname{Pol}_n(p)$ for its diagonal
 $p(z) = P(z, \dotsc, z)$, and $P$ is stable if and only if $p$ has no zeros
 in the open upper half-plane. If $P$ has real coefficients, then $P$ is real
@@ -174,7 +184,7 @@ multiaffine polynomial.
 **Theorem.** The blockwise polarization of $P$ is stable if and only if $P$
 is stable.
 
-**Theorem** (Lieb–Sokal, multiaffine case). Let $F$ and $G$ be stable
+**Theorem** (Lieb–Sokal, multiaffine case). [@LiebSokal1981] Let $F$ and $G$ be stable
 multiaffine polynomials in $\mathbb{C}[z_1, \dotsc, z_m]$. Then
 $F(-\partial)\, G$, the result of substituting $-\partial/\partial z_i$ for
 $z_i$ in $F$ and applying it to $G$, is zero or stable.
@@ -211,24 +221,11 @@ operation that preserves stability of multiaffine polynomials.
 
 ## References
 
-J. H. Grace, “The zeros of a polynomial,” *Proc. Cambridge Philos. Soc.* 11
-(1902), 352–357; J. L. Walsh, “On the location of the roots of certain types
-of polynomials,” *Trans. Amer. Math. Soc.* 24 (1922), 163–180;
-G. Szegő, “Bemerkungen zu einem Satz von J. H. Grace über die
-Wurzeln algebraischer Gleichungen,” *Math. Z.* 13 (1922), 28–55; Q. I. Rahman
-and G. Schmeisser, *Analytic Theory of Polynomials*, Oxford University Press,
-2002; E. H. Lieb and A. D. Sokal, “A general Lee–Yang theorem for
-one-component and multicomponent ferromagnets,” *Comm. Math. Phys.* 80 (1981),
-153–179, [doi:10.1007/bf01213009](https://doi.org/10.1007/bf01213009);
-J. Borcea and P. Brändén, “The Lee–Yang and Pólya–Schur programs. II. Theory
-of stable polynomials and applications,” *Comm. Pure Appl. Math.* 62 (2009),
-1595–1631, [doi:10.1002/cpa.20295](https://doi.org/10.1002/cpa.20295);
-A. Gribinski and A. W. Marcus, “A rectangular additive
-convolution for polynomials,” *Combinatorial Theory* 2(1) (2022), #16,
-Theorem 2.4, [doi:10.5070/c62156888](https://doi.org/10.5070/c62156888). See
-[apolar polynomials](https://www.symmetricfunctions.com/stablePolynomials.htm#apolarPolynomial)
-and [polarization](https://www.symmetricfunctions.com/stablePolynomials.htm#polarizationDefinition)
-on symmetricfunctions.com.
+The classical sources are [@Grace1902; @Walsh1922; @Szego1922] and
+[@RahmanSchmeisser2002]. See also [@LiebSokal1981; @BorceaBranden2009II;
+@GribinskiMarcus2022, Thm. 2.4], as well as
+[apolar polynomials](sf:apolarPolynomial) and
+[polarization](sf:polarizationDefinition) on symmetricfunctions.com.
 <!-- /realrooted-catalog-content -->
 
 This module exposes Grace's theorem for closed disks and closed upper
diff --git a/RealRooted/Challenges/Hadamard.lean b/RealRooted/Challenges/Hadamard.lean
index cf866b96f..201a4187d 100644
--- a/RealRooted/Challenges/Hadamard.lean
+++ b/RealRooted/Challenges/Hadamard.lean
@@ -19,7 +19,17 @@ version = 1
 section = "theorems"
 slug = "hadamard-products"
 formalized = "2026-07-03"
-authors = ["Maló", "Pólya", "Schur", "Brenti", "Wagner", "Garloff", "Kostov", "Shapiro", "Brändén",
+references = [
+  "SchurPolya1914",
+  "Wagner1992",
+  "GarloffWagner1996",
+  "BrandenFerroniJochemko2024x",
+  "LiuMao2026Hadamard",
+  "KostovShapiro2006",
+  "Kostov2010",
+]
+authors = [
+  "Maló", "Pólya", "Schur", "Brenti", "Wagner", "Garloff", "Kostov", "Shapiro", "Brändén",
   "Ferroni", "Jochemko", "Liu", "Mao"]
 years = [1895, 1914, 1989, 1992, 1996, 2006, 2010, 2024, 2026]
 
@@ -143,7 +153,8 @@ Pólya frequency sequences is again a Pólya frequency sequence. Equivalently,
 the entrywise product of their totally nonnegative Toeplitz matrices is
 totally nonnegative.
 
-**Theorem** (Kostov–Shapiro; Kostov). Let $f$ and $g$ have degree $n$, let
+**Theorem** (Kostov–Shapiro; Kostov). [@KostovShapiro2006; @Kostov2010] Let
+$f$ and $g$ have degree $n$, let
 $a \neq 0$ be a root of $f$ of multiplicity $m$ and $b \neq 0$ a root of $g$ of
 multiplicity $l$, with $m + l \geq n$. Then $-ab$ is a root of the Schur–Szegő
 composition $f *_n g$ of multiplicity exactly $m + l - n$; in particular it is not a
@@ -192,11 +203,12 @@ nonnegative coefficients):
   factorial product), with coefficients $k!\, a_k b_k$, preserves interlacing in
   its first argument.
 
-**Theorem (Schur).** If $f = \sum_k a_k x^k$ is real-rooted and
+**Theorem (Schur).** [@SchurPolya1914] If $f = \sum_k a_k x^k$ is real-rooted and
 $g = \sum_k b_k x^k$ is real-rooted with all zeros of one sign, then the
 factorial Hadamard product $\sum_k k!\, a_k b_k x^k$ is zero or real-rooted.
 
-**Theorem (Liu–Mao).** For a polynomial $p$ of degree $d$ let $W(p)$ be its Euler
+**Theorem (Liu–Mao).** [@LiuMao2026Hadamard] For a polynomial $p$ of degree
+$d$ let $W(p)$ be its Euler
 numerator, $\sum_{n \geq 0} p(n) x^n = W(p)(x) / (1-x)^{d+1}$. If the coefficients of
 $W(p)$ and $W(q)$ are nonnegative, log-concave and without internal zeros, then so are
 the coefficients of $W(pq)$. The no-internal-zeros hypothesis cannot be dropped:
@@ -205,37 +217,27 @@ $W(p) = 1 + x^3$ and $W(q) = x$ give $W(pq) = 4x + 3x^3 + x^4$.
 ## References
 
 E. Maló, “Note sur les équations algébriques dont toutes les racines sont
-réelles,” *Journal de Mathématiques Spéciales* 4 (1895), 7–10; G. Pólya and
-I. Schur, “Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen
-Gleichungen,” *Journal für die reine und angewandte Mathematik* 144 (1914),
-89–113; D. G. Wagner, “Total positivity of Hadamard products,” *Journal of
-Mathematical Analysis and Applications* 163 (1992), 459–483; J. Garloff and
-D. G. Wagner, “Hadamard products of stable polynomials are stable,” *Journal
-of Mathematical Analysis and Applications* 202 (1996), 797–809; P. Brändén,
-L. Ferroni and K. Jochemko, “Preservation of inequalities under Hadamard
-products,” arXiv:2408.12386 (2024); Y. Liu and J. Mao, “Preservation of
-log-concavity under Hadamard products,” arXiv:2609.11589 (2026); V. Kostov and B.
-Shapiro, “On the Schur–Szegő composition of polynomials,” *C. R. Math. Acad. Sci.
-Paris* 343 (2006), 81–86; V. P. Kostov, “Interlacing properties and the Schur–Szegő
-composition,” *Functional Analysis and Other Mathematics* 3 (2010), 65–74; V. P. Kostov,
-“Eigenvectors in the context of the Schur–Szegő composition of polynomials,” *Mathematica
-Balkanica* 22 (2008), 155–173.  See the
-[Hadamard-product overview](https://www.symmetricfunctions.com/realRooted.htm#hadamardProductTheorems),
-[Schur–Szegő composition](https://www.symmetricfunctions.com/realRooted.htm#schurSzegoComposition),
-and the [finite multiplier criterion](https://www.symmetricfunctions.com/realRooted.htm#finiteMultiplierSequenceCriterion)
+réelles,” *Journal de Mathématiques Spéciales* 4 (1895), 7–10. The matched
+sources are [@SchurPolya1914; @Wagner1992; @GarloffWagner1996;
+@BrandenFerroniJochemko2024x; @LiuMao2026Hadamard; @KostovShapiro2006;
+@Kostov2010]. V. P. Kostov, “Eigenvectors in the context of the Schur–Szegő
+composition of polynomials,” *Mathematica Balkanica* 22 (2008), 155–173. See the
+[Hadamard-product overview](sf:hadamardProductTheorems),
+[Schur–Szegő composition](sf:schurSzegoComposition), and the
+[finite multiplier criterion](sf:finiteMultiplierSequenceCriterion)
 on symmetricfunctions.com.
 <!-- /realrooted-catalog-content -->
 
 Human statements:
 
 * Hadamard product theorems:
-  https://www.symmetricfunctions.com/realRooted.htm#hadamardProductTheorems
+  sf:hadamardProductTheorems
 * Schur--Szego composition:
-  https://www.symmetricfunctions.com/realRooted.htm#schurSzegoComposition
+  sf:schurSzegoComposition
 * Polya-frequency sequences:
-  https://www.symmetricfunctions.com/polyaFrequency.htm#aissenSchoenbergWhitney
+  sf:aissenSchoenbergWhitney
 * Finite multiplier criterion:
-  https://www.symmetricfunctions.com/realRooted.htm#finiteMultiplierSequenceCriterion
+  sf:finiteMultiplierSequenceCriterion
 
 Original publications include:
 
diff --git a/RealRooted/Challenges/HarperCLT.lean b/RealRooted/Challenges/HarperCLT.lean
index 1d958b7b9..4225dbdf5 100644
--- a/RealRooted/Challenges/HarperCLT.lean
+++ b/RealRooted/Challenges/HarperCLT.lean
@@ -14,6 +14,11 @@ version = 1
 section = "theorems"
 slug = "harper-clt"
 formalized = "2026-10-12"
+references = [
+  "Bender1973",
+  "Bona2009",
+  "Harper1967",
+]
 authors = ["Harper", "Bender", "Goncharov", "David", "Barton", "Bóna"]
 years = [1944, 1962, 1967, 1973, 2009]
 
@@ -115,7 +120,7 @@ and $P_n''(1)$ from forced scalar recurrences and writes the limits algebraicall
 $b(1)$ and their derivatives. With real-rootedness of the rows and $\sigma^2(P_n) \to \infty$,
 Harper's theorem then gives asymptotic normality.
 
-**Theorem** (Harper; Bender). Let $P_n$ be real-rooted polynomials with
+**Theorem** (Harper; Bender). [@Harper1967; @Bender1973] Let $P_n$ be real-rooted polynomials with
 nonnegative coefficients and $\sigma^2(P_n) \to \infty$. Then the standardized
 coefficient distributions, the laws of $(K_n - \mu_n)/\sigma_n$, converge weakly
 to the standard normal distribution.
@@ -168,18 +173,10 @@ turns pointwise convergence of characteristic functions into weak convergence.
 
 ## References
 
-L. H. Harper, “Stirling behavior is asymptotically normal,” *Annals of
-Mathematical Statistics* 38 (1967), 410–414,
-[doi:10.1214/aoms/1177698956](https://doi.org/10.1214/aoms/1177698956);
-E. A. Bender, “Central and local limit theorems applied to asymptotic
-enumeration,” *Journal of Combinatorial Theory, Series A* 15 (1973), 91–111,
-[doi:10.1016/0097-3165(73)90038-1](https://doi.org/10.1016/0097-3165(73)90038-1);
-V. L. Goncharov, “Some facts from
-combinatorics,” *Izv. Akad. Nauk SSSR Ser. Mat.* 8 (1944), 3–48; F. N. David
-and D. E. Barton, *Combinatorial Chance*, Griffin, London, 1962; M. Bóna, “Real
-zeros and normal distribution for statistics on Stirling permutations defined
-by Gessel and Stanley,” *SIAM Journal on Discrete Mathematics* 23 (2008/09),
-401–406, [doi:10.1137/070702254](https://doi.org/10.1137/070702254).
+See [@Harper1967; @Bender1973]. V. L. Goncharov, “Some facts from combinatorics,”
+*Izv. Akad. Nauk SSSR Ser. Mat.* 8 (1944), 3–48. F. N. David and D. E. Barton,
+*Combinatorial Chance*, Griffin, London, 1962. For Stirling permutations,
+see [@Bona2009].
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in
diff --git a/RealRooted/Challenges/HeilmannLieb.lean b/RealRooted/Challenges/HeilmannLieb.lean
index f1b4c27f0..940010e62 100644
--- a/RealRooted/Challenges/HeilmannLieb.lean
+++ b/RealRooted/Challenges/HeilmannLieb.lean
@@ -10,6 +10,11 @@ version = 1
 section = "theorems"
 slug = "heilmann-lieb"
 formalized = "2026-10-06"
+references = [
+  "HeilmannLieb1972",
+  "Engstrom2007",
+  "ChudnovskySeymour2007",
+]
 authors = ["Heilmann", "Lieb", "Godsil", "Engström"]
 years = [1972, 1981, 2007]
 
@@ -77,7 +82,8 @@ where $M$ ranges over the matchings of $G$, that is, the sets of pairwise
 disjoint edges. Given edge weights $w \colon E(G) \to \mathbb{R}$, the
 weighted version replaces $x^{|M|}$ by $\prod_{e \in M} w(e)\, x^{|M|}$.
 
-**Theorem** (Heilmann–Lieb). The polynomial $m_G(x)$ is real-rooted. More
+**Theorem** (Heilmann–Lieb). [@HeilmannLieb1972, Thm. 4.2] The polynomial
+$m_G(x)$ is real-rooted. More
 generally, if all edge weights are nonnegative, then the weighted
 matching-generating polynomial is real-rooted with nonnegative coefficients
 and nonpositive zeros, so its coefficients form a Pólya frequency sequence.
@@ -98,7 +104,7 @@ most $\Delta \geq 2$, every zero $x$ of $\mu_G$ satisfies
 $|x| \leq 2\sqrt{\Delta - 1}$. Heilmann and Lieb prove the strict inequality;
 the Lean statement records the weak form.
 
-**Theorem** (Engström). Let $G$ be a finite claw-free graph and let
+**Theorem** (Engström). [@Engstrom2007, Thm. 2.5] Let $G$ be a finite claw-free graph and let
 $w \colon V(G) \to \mathbb{R}$ be nonnegative vertex weights. Then the
 weighted independence polynomial
 $$
@@ -139,19 +145,11 @@ whenever $v$ has at most $\Delta - 1$ neighbours, and $\mu_G$ is even or odd.
 
 ## References
 
-O. J. Heilmann and E. H. Lieb, “Theory of monomer-dimer systems,” *Comm.
-Math. Phys.* 25 (1972), 190–232,
-[doi:10.1007/bf01877590](https://doi.org/10.1007/bf01877590), Theorem 4.2;
-C. D. Godsil, “Matchings and
-walks in graphs,” *J. Graph Theory* 5 (1981), 285–297; A. Engström, “Inequalities on
-well-distributed point sets on circles,” *J. Inequal. Pure Appl. Math.* 8
-(2007), Theorem 2.5; M. Chudnovsky and P. Seymour, “The roots of the
-independence polynomial of a clawfree graph,” *J. Combin. Theory Ser. B* 97
-(2007), 350–357, [doi:10.1016/j.jctb.2006.06.001](https://doi.org/10.1016/j.jctb.2006.06.001).
-See the
-[matching polynomial](https://www.symmetricfunctions.com/realRootedGraphs.htm#matchingPolynomial)
-and the
-[weighted independence polynomial](https://www.symmetricfunctions.com/realRootedGraphs.htm#weightedIndependencePolynomial)
+See [@HeilmannLieb1972, Thm. 4.2]. C. D. Godsil, “Matchings and walks in
+graphs,” *J. Graph Theory* 5 (1981), 285–297. The claw-free specialization is
+[@Engstrom2007, Thm. 2.5]; see also [@ChudnovskySeymour2007], the
+[matching polynomial](sf:matchingPolynomial), and the
+[weighted independence polynomial](sf:weightedIndependencePolynomial)
 on symmetricfunctions.com.
 <!-- /realrooted-catalog-content -->
 
diff --git a/RealRooted/Challenges/HermiteBiehlerHurwitz.lean b/RealRooted/Challenges/HermiteBiehlerHurwitz.lean
index 994e7c2b6..5736d2c06 100644
--- a/RealRooted/Challenges/HermiteBiehlerHurwitz.lean
+++ b/RealRooted/Challenges/HermiteBiehlerHurwitz.lean
@@ -17,6 +17,9 @@ version = 1
 section = "theorems"
 slug = "hermite-biehler-hurwitz"
 formalized = "2026-07-10"
+references = [
+  "Holtz2003",
+]
 authors = ["Holtz"]
 years = [2003]
 
@@ -129,27 +132,22 @@ the minors along the Routh reduction.
 
 ## References
 
-O. Holtz, “Hermite–Biehler, Routh–Hurwitz, and total positivity,” *Linear
-Algebra and its Applications* 372 (2003), 105–110,
-[doi:10.1016/s0024-3795(03)00501-9](https://doi.org/10.1016/s0024-3795(03)00501-9).
-See also the
-[Hermite–Biehler theorem](https://www.symmetricfunctions.com/realRootedInterlacing.htm#hermiteBiehlerTheorem)
-and the [coefficient signs of Hurwitz-stable polynomials](https://www.symmetricfunctions.com/realRootedInterlacing.htm#hurwitzImpliesCoefficientPositivity)
-on symmetricfunctions.com. The related
-[Hurwitz–Lace criterion](https://www.symmetricfunctions.com/realRootedInterlacing.htm#hurwitzLaceCriterion)
-there is stated for a different matrix, built from two polynomials, and is not
-formalized here.
+See [@Holtz2003], the
+[Hermite–Biehler theorem](sf:hermiteBiehlerTheorem), and the
+[coefficient signs of Hurwitz-stable polynomials](sf:hurwitzImpliesCoefficientPositivity).
+The related [Hurwitz–Lace criterion](sf:hurwitzLaceCriterion) is stated for a
+different matrix, built from two polynomials, and is not formalized here.
 <!-- /realrooted-catalog-content -->
 
 Human statements:
 
 * Hermite--Biehler:
-  https://www.symmetricfunctions.com/realRootedInterlacing.htm#hermiteBiehlerTheorem
+  sf:hermiteBiehlerTheorem
 * Hurwitz stability implies coefficient positivity:
-  https://www.symmetricfunctions.com/realRootedInterlacing.htm#hurwitzImpliesCoefficientPositivity
+  sf:hurwitzImpliesCoefficientPositivity
 * Related, not formalized: the Hurwitz--Lace criterion, stated for a different
   matrix:
-  https://www.symmetricfunctions.com/realRootedInterlacing.htm#hurwitzLaceCriterion
+  sf:hurwitzLaceCriterion
 
 Classical references include C. Hermite, A. Hurwitz, M. G. Krein, M. A. Naimark,
 and the modern account by O. Holtz, "Hermite-Biehler, Routh-Hurwitz, and
diff --git a/RealRooted/Challenges/HermitePolynomials.lean b/RealRooted/Challenges/HermitePolynomials.lean
index f124408eb..74765df1f 100644
--- a/RealRooted/Challenges/HermitePolynomials.lean
+++ b/RealRooted/Challenges/HermitePolynomials.lean
@@ -9,6 +9,9 @@ version = 1
 section = "families"
 slug = "hermite-polynomials"
 formalized = "2026-09-07"
+references = [
+  "Szego1939",
+]
 
 [[definitions]]
 name = "RealRooted.hermiteReal"
@@ -76,8 +79,7 @@ positive.
 
 ## References
 
-G. Szegő, *Orthogonal Polynomials*, American Mathematical Society Colloquium
-Publications 23 (1939).
+See [@Szego1939].
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in `RealRooted.Hermite`.
diff --git a/RealRooted/Challenges/HermitePoulain.lean b/RealRooted/Challenges/HermitePoulain.lean
index ce9ec8489..a3764cdb9 100644
--- a/RealRooted/Challenges/HermitePoulain.lean
+++ b/RealRooted/Challenges/HermitePoulain.lean
@@ -41,13 +41,13 @@ the two orientations, where either image may be zero.
 ## References
 
 The theorem goes back to Hermite and Poulain; see the
-[Hermite–Poulain theorem on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedInterlacing.htm#hermitePoulainTheorem)
+[Hermite–Poulain theorem on symmetricfunctions.com](sf:hermitePoulainTheorem)
 for further references, and the
-[interlacing consequence](https://www.symmetricfunctions.com/realRootedInterlacing.htm#hermitePoulainInterlacing).
+[interlacing consequence](sf:hermitePoulainInterlacing).
 <!-- /realrooted-catalog-content -->
 
 Human statement:
-https://www.symmetricfunctions.com/realRootedInterlacing.htm#hermitePoulainTheorem
+sf:hermitePoulainTheorem
 
 Original references include C. Hermite, G. Polya--I. Schur, N. Obreschkoff,
 and B. Ya. Levin's account of entire functions.
diff --git a/RealRooted/Challenges/HosterStump.lean b/RealRooted/Challenges/HosterStump.lean
index 992d5fbfa..9fdfd226b 100644
--- a/RealRooted/Challenges/HosterStump.lean
+++ b/RealRooted/Challenges/HosterStump.lean
@@ -8,6 +8,9 @@ version = 1
 section = "theorems"
 slug = "chow-polynomials-simplicial-posets"
 formalized = "2026-10-08"
+references = [
+  "HosterStump2025x",
+]
 authors = ["Hoster", "Stump"]
 years = [2025]
 
@@ -73,7 +76,7 @@ H^{\mathrm{aug}}_{\hat P}(x) = \sum_{\substack{S \subseteq \{1,\dots,n\}\\ S \te
 $$
 where a set is *isolated* if it contains no two consecutive integers.
 
-**Theorem** (Hoster–Stump). If the $h$-vector of $P$ is nonnegative, then
+**Theorem** (Hoster–Stump). [@HosterStump2025x] If the $h$-vector of $P$ is nonnegative, then
 $H_{\hat P}$, $H_{\hat P^*}$ and $H^{\mathrm{aug}}_{\hat P}$ are real-rooted, and the
 roots of $H_{\hat P^*}$ interlace those of $H^{\mathrm{aug}}_{\hat P}$.
 
@@ -96,10 +99,9 @@ false; the library records an explicit counterexample.
 
 ## References
 
-E. Hoster and C. Stump, “Chow polynomials of simplicial posets with positive $h$-vector are
-real-rooted,” arXiv:2508.15538 (2025); R. P. Stanley, *Combinatorics and Commutative
-Algebra*, 2nd ed., Birkhäuser, 1996, III.6; L. Ferroni, J. P. Matherne and L. Vecchi, “Chow
-functions for partially ordered sets,” 2024.
+See [@HosterStump2025x]. R. P. Stanley, *Combinatorics and Commutative
+Algebra*, 2nd ed., Birkhäuser, 1996, III.6. L. Ferroni, J. P. Matherne and
+L. Vecchi, “Chow functions for partially ordered sets,” 2024.
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in `RealRooted.HosterStump`.
diff --git a/RealRooted/Challenges/Hutchinson.lean b/RealRooted/Challenges/Hutchinson.lean
index b9d3427cb..b64ac4dfe 100644
--- a/RealRooted/Challenges/Hutchinson.lean
+++ b/RealRooted/Challenges/Hutchinson.lean
@@ -8,6 +8,10 @@ version = 1
 section = "theorems"
 slug = "hutchinson"
 formalized = "2026-10-07"
+references = [
+  "Hutchinson1923",
+  "Kurtz1992",
+]
 authors = ["Hutchinson"]
 years = [1923]
 
@@ -26,7 +30,8 @@ label = "The Laguerre–Pólya class is closed under locally uniform limits"
 <!-- realrooted-catalog-content -->
 # Hutchinson's theorem
 
-**Theorem** (Hutchinson). If $a_k > 0$ and $a_{k+1}^2 \geq 4 a_k a_{k+2}$ for
+**Theorem** (Hutchinson). [@Hutchinson1923] If $a_k > 0$ and
+$a_{k+1}^2 \geq 4 a_k a_{k+2}$ for
 all $k$, then $\sum_k a_k z^k$ is an entire function in the Laguerre–Pólya
 class: a locally uniform limit of real-rooted real polynomials.
 
@@ -44,12 +49,7 @@ Laguerre–Pólya class is closed under locally uniform limits.
 
 ## References
 
-J. I. Hutchinson, “On a remarkable class of entire functions,” *Transactions
-of the American Mathematical Society* 25 (1923), 325–332,
-[doi:10.1090/s0002-9947-1923-1501248-1](https://doi.org/10.1090/s0002-9947-1923-1501248-1);
-D. C. Kurtz, “A sufficient condition for all the roots of a polynomial to be
-real,” *American Mathematical Monthly* 99 (1992), 259–263,
-[doi:10.1080/00029890.1992.11995845](https://doi.org/10.1080/00029890.1992.11995845).
+See [@Hutchinson1923; @Kurtz1992].
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in `RealRooted.Hutchinson`
diff --git a/RealRooted/Challenges/Interlacing.lean b/RealRooted/Challenges/Interlacing.lean
index 4c58ff6e7..351f1f25f 100644
--- a/RealRooted/Challenges/Interlacing.lean
+++ b/RealRooted/Challenges/Interlacing.lean
@@ -8,6 +8,9 @@ version = 1
 section = "concepts"
 slug = "interlacing"
 formalized = "2026-10-05"
+references = [
+  "Fisk2006x",
+]
 authors = ["Fisk"]
 years = [2006]
 
@@ -69,10 +72,8 @@ $\deg g = \deg f + 1$.
 
 ## References
 
-Steve Fisk, [“Polynomials, roots, and
-interlacing,”](https://arxiv.org/abs/math/0612833) arXiv:math/0612833 (2006).
-See also the
-[interlacing overview on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedInterlacing.htm#interlacesPolynomial).
+See [@Fisk2006x] and the
+[interlacing overview on symmetricfunctions.com](sf:interlacesPolynomial).
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/InterlacingClosure.lean b/RealRooted/Challenges/InterlacingClosure.lean
index 4653316fa..0c16b056e 100644
--- a/RealRooted/Challenges/InterlacingClosure.lean
+++ b/RealRooted/Challenges/InterlacingClosure.lean
@@ -13,6 +13,11 @@ version = 1
 section = "theorems"
 slug = "interlacing-closure"
 formalized = "2026-10-06"
+references = [
+  "Fisk2006x",
+  "Branden2015",
+  "BrandenLeite2024x",
+]
 authors = ["Fisk", "Brändén", "Saud Leite", "Hlavacek", "Solus"]
 years = [2006, 2015, 2021, 2024]
 
@@ -58,7 +63,8 @@ side is also allowed to be zero.
 coefficients. If $(sx + t) f + g$ is real-rooted for all $s, t > 0$, then
 $f \ll g$.
 
-**Cone lemma (Brändén, Saud Leite).** Let $f_0, \dots, f_n$ be real-rooted with
+**Cone lemma (Brändén, Saud Leite).** [@BrandenLeite2024x, Lemma 3.1] Let
+$f_0, \dots, f_n$ be real-rooted with
 nonnegative coefficients, and suppose $f_i \ll f_j$ for all $i < j \le n$.
 Then for all $\lambda_0, \dots, \lambda_n \ge 0$ the combination
 $F = \sum_i \lambda_i f_i$ satisfies $f_0 \ll F \ll f_n$, where either side
@@ -90,17 +96,9 @@ turn this into $f \ll g$ by Wagner's lemma.
 
 ## References
 
-S. Fisk, *Polynomials, roots, and interlacing*, arXiv:math/0612833 (2006);
-P. Brändén, “Unimodality, log-concavity, real-rootedness and beyond,”
-*Handbook of Enumerative Combinatorics*, CRC Press, 2015, Lemma 7.8.4;
-P. Brändén and L. Saud Maia Leite, “Totally nonnegative matrices, chain
-enumeration and zeros of polynomials,” *Advances in Mathematics* 487 (2026),
-110760, [doi:10.1016/j.aim.2025.110760](https://doi.org/10.1016/j.aim.2025.110760),
-arXiv:2412.06595, Lemma 3.1.
-See also the
-[interlacing sequences on symmetricfunctions.com][seq].
-
-[seq]: https://www.symmetricfunctions.com/realRootedInterlacing.htm#interlacingSequenceDefinition
+See [@Fisk2006x], [@Branden2015, Lemma 7.8.4], and
+[@BrandenLeite2024x, Lemma 3.1]. See also the
+[interlacing sequences on symmetricfunctions.com](sf:interlacingSequenceDefinition).
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade. The proofs live in `RealRooted.AffineFamily`,
diff --git a/RealRooted/Challenges/IteratedDerivativeShift.lean b/RealRooted/Challenges/IteratedDerivativeShift.lean
index eff8291ce..32eae6d35 100644
--- a/RealRooted/Challenges/IteratedDerivativeShift.lean
+++ b/RealRooted/Challenges/IteratedDerivativeShift.lean
@@ -10,6 +10,10 @@ version = 1
 section = "theorems"
 slug = "iterated-derivative-shift"
 formalized = "2026-10-05"
+references = [
+  "SchurPolya1914",
+  "Obreschkoff1963",
+]
 
 [[theorems]]
 name = "RealRooted.splits_tDeriv"
@@ -129,12 +133,7 @@ split; the factor may occur on either side of the iterate.
 
 ## References
 
-That `p ↦ p - ε p'` preserves real-rootedness is a classical fact going back to Laguerre; for the
-general theory of such preservers see G. Pólya and J. Schur, “Über zwei Arten von Faktorenfolgen in
-der Theorie der algebraischen Gleichungen,” *J. Reine Angew. Math.* 144 (1914), 89–113. The
-iterated regularization is used in the converse of Obreschkoff's theorem: N. Obreschkoff,
-*Verteilung und Berechnung der Nullstellen reeller Polynome*, VEB Deutscher Verlag der
-Wissenschaften, 1963.
+See [@SchurPolya1914; @Obreschkoff1963].
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/JacobiDeformation.lean b/RealRooted/Challenges/JacobiDeformation.lean
index 13f161e0f..c8215ec1f 100644
--- a/RealRooted/Challenges/JacobiDeformation.lean
+++ b/RealRooted/Challenges/JacobiDeformation.lean
@@ -10,6 +10,9 @@ version = 1
 section = "families"
 slug = "jacobi-deformation"
 formalized = "2026-10-06"
+references = [
+  "MicchelliWilloughby1979",
+]
 authors = ["Micchelli", "Willoughby", "Alexandersson"]
 years = [1979, 2026]
 
@@ -105,7 +108,8 @@ eigenvalues of the Jacobi operator.
 2. At the roots of a quasi-Jacobi polynomial $p_q - \tau p_{q-1}$, exact
    quadrature symmetrizes the collocation matrix of the Jacobi operator, and
    all its entries are strictly positive.
-3. The **Micchelli–Willoughby theorem** makes the initial spectral products of
+3. The **Micchelli–Willoughby theorem** [@MicchelliWilloughby1979] makes the initial
+   spectral products of
    that matrix entrywise nonnegative: if $A$ is a real symmetric matrix with
    nonnegative entries and, in the simple-spectrum case formalized here,
    eigenvalues $\mu_1 < \dots < \mu_N$, then
@@ -121,15 +125,11 @@ eigenvalues of the Jacobi operator.
 
 ## References
 
-The theorem and its proof are original to this library (P. Alexandersson,
-2026); there is no preprint yet.
-
-C. A. Micchelli and R. A. Willoughby, “On functions which preserve the class
-of Stieltjes matrices,” *Linear Algebra and its Applications* 23 (1979),
-141–156, [doi:10.1016/0024-3795(79)90098-3](https://doi.org/10.1016/0024-3795(79)90098-3).
+The theorem and its proof are original to this library (P. Alexandersson, 2026);
+there is no preprint yet. For Stieltjes matrices, see [@MicchelliWilloughby1979].
 The triangle is [OEIS A132885](https://oeis.org/A132885); the central trinomial
-coefficients are [OEIS A002426](https://oeis.org/A002426). The Jacobi
-polynomials used in the proof are on the
+coefficients are [OEIS A002426](https://oeis.org/A002426). The Jacobi polynomials
+used in the proof are on the
 [Jacobi polynomials page](/RealRooted/families/jacobi-polynomials/).
 <!-- /realrooted-catalog-content -->
 
diff --git a/RealRooted/Challenges/JacobiPolynomials.lean b/RealRooted/Challenges/JacobiPolynomials.lean
index 960ae2836..44c93807b 100644
--- a/RealRooted/Challenges/JacobiPolynomials.lean
+++ b/RealRooted/Challenges/JacobiPolynomials.lean
@@ -14,6 +14,9 @@ version = 1
 section = "families"
 slug = "jacobi-polynomials"
 formalized = "2026-09-07"
+references = [
+  "Szego1939",
+]
 authors = ["Jacobi", "Markoff", "Szegő"]
 years = [1859, 1886, 1939]
 
@@ -145,13 +148,10 @@ proved directly.
 ## References
 
 C. G. J. Jacobi, “Untersuchungen über die Differentialgleichung der
-hypergeometrischen Reihe,” *Journal für die reine und angewandte Mathematik*
-56 (1859), 149–165;
-A. Markoff, “Sur les racines de certaines équations (seconde note),”
-*Mathematische Annalen* 27 (1886), 177–182;
-G. Szegő, *Orthogonal Polynomials*, American Mathematical Society Colloquium
-Publications 23 (1939), Chapters IV and VI.
-The general three-term theory is on the
+hypergeometrischen Reihe,” *Journal für die reine und angewandte Mathematik* 56
+(1859), 149–165; A. Markoff, “Sur les racines de certaines équations (seconde
+note),” *Mathematische Annalen* 27 (1886), 177–182. See also [@Szego1939,
+Chs. IV and VI]. The general three-term theory is on the
 [Favard page](/RealRooted/theorems/favard/).
 <!-- /realrooted-catalog-content -->
 
diff --git a/RealRooted/Challenges/JacobiStirlingDescent.lean b/RealRooted/Challenges/JacobiStirlingDescent.lean
index 20694d185..e1ff18deb 100644
--- a/RealRooted/Challenges/JacobiStirlingDescent.lean
+++ b/RealRooted/Challenges/JacobiStirlingDescent.lean
@@ -8,6 +8,10 @@ version = 1
 section = "families"
 slug = "jacobi-stirling-descent"
 formalized = "2026-10-07"
+references = [
+  "GesselLinZeng2012",
+  "MaWang2026x",
+]
 authors = ["Gessel", "Lin", "Zeng", "Ma", "Wang"]
 years = [2012, 2026]
 
@@ -71,7 +75,8 @@ $$D_L f = (1 + Lx) f + x(1 - x) f'.$$
 Here these recurrences define the polynomials. The permutation model is not
 formalized.
 
-**Theorem** (Ma–Wang). Let $1 \le i \le k$ with $i \in \{1, 2, 3, k-2, k-1\}$.
+**Theorem** (Ma–Wang). [@MaWang2026x, Sec. 4] Let $1 \le i \le k$ with
+$i \in \{1, 2, 3, k-2, k-1\}$.
 Every nonzero nonnegative combination $\sum_{|S| = i} c_S A_{k,S}$ has only
 simple negative zeros. In particular, $A_{k,i}$ is real-rooted.
 
@@ -95,12 +100,7 @@ all strictly interlace a common polynomial keep that property.
 
 ## References
 
-I. M. Gessel, Z. Lin and J. Zeng, “Jacobi–Stirling polynomials and
-$P$-partitions,” *European Journal of Combinatorics* 33 (2012), 1987–2000,
-[doi:10.1016/j.ejc.2012.06.008](https://doi.org/10.1016/j.ejc.2012.06.008);
-S.-M. Ma and M.-X. Wang, “Binomial expansions of Jacobi–Stirling numbers and
-real-rootedness of Jacobi–Stirling descent polynomials,” arXiv:2610.03111
-(2026), Section 4.
+See [@GesselLinZeng2012] and [@MaWang2026x, Sec. 4].
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in
diff --git a/RealRooted/Challenges/JacobiStirlingNumbers.lean b/RealRooted/Challenges/JacobiStirlingNumbers.lean
index 542191b5e..e3ea9de4e 100644
--- a/RealRooted/Challenges/JacobiStirlingNumbers.lean
+++ b/RealRooted/Challenges/JacobiStirlingNumbers.lean
@@ -10,6 +10,11 @@ version = 1
 section = "families"
 slug = "jacobi-stirling-numbers"
 formalized = "2026-09-13"
+references = [
+  "GelineauZeng2010",
+  "Mongelli2012",
+  "AndrewsEggeGawronskiLittlejohn2013",
+]
 authors = ["Gelineau", "Zeng", "Mongelli"]
 years = [2010, 2012]
 
@@ -66,7 +71,7 @@ symmetric function and $\mathrm{JS}(n,k)$ a complete homogeneous symmetric
 function of the weights $i(i+z)$. The signed first-kind triangle and the
 second-kind triangle are mutually inverse.
 
-**Theorem** (Mongelli). For $z \geq -1$ the first-kind triangle
+**Theorem** (Mongelli). [@Mongelli2012] For $z \geq -1$ the first-kind triangle
 $(\mathrm{Jc}(n,k;z))_{n,k}$ is totally nonnegative, and every row is a
 Pólya-frequency sequence: the row polynomial $\sum_k \mathrm{Jc}(n,k;z)\,x^k$
 factors into linear terms $x + i(i+z)$ with nonnegative constants.
@@ -80,14 +85,7 @@ network is needed.
 
 ## References
 
-Y. Gelineau and J. Zeng, [“Combinatorial interpretations of the Jacobi–Stirling
-numbers,”](https://doi.org/10.37236/342) *Electronic Journal of Combinatorics*
-17 (2010), #R70; P. Mongelli, [“Total positivity properties of Jacobi–Stirling
-numbers,”](https://doi.org/10.1016/j.aam.2011.06.008) *Advances in Applied
-Mathematics* 48 (2012), 354–364; G. E. Andrews, E. S. Egge, W. Gawronski and
-L. L. Littlejohn, [“The Jacobi–Stirling numbers,”](https://doi.org/10.1016/j.jcta.2012.08.006)
-*Journal of Combinatorial Theory, Series A* 120 (2013), 288–303.
-
+See [@GelineauZeng2010; @Mongelli2012; @AndrewsEggeGawronskiLittlejohn2013].
 The descent polynomials of Jacobi–Stirling permutations are on the
 [Jacobi–Stirling descent page](/RealRooted/families/jacobi-stirling-descent/).
 <!-- /realrooted-catalog-content -->
diff --git a/RealRooted/Challenges/KostovEigenpolynomials.lean b/RealRooted/Challenges/KostovEigenpolynomials.lean
index a4f4f167f..b0c093e04 100644
--- a/RealRooted/Challenges/KostovEigenpolynomials.lean
+++ b/RealRooted/Challenges/KostovEigenpolynomials.lean
@@ -10,6 +10,9 @@ version = 1
 section = "theorems"
 slug = "kostov-eigenpolynomials"
 formalized = "2026-10-12"
+references = [
+  "Kostov2010",
+]
 authors = ["Kostov"]
 years = [2015]
 
@@ -63,17 +66,16 @@ through the reduced eigenpolynomials $Q_{n,j}$: each $Q_{n,j}$ is monic of degre
 $j$, satisfies $X^j Q_{n,j}(1/X) = (-1)^j Q_{n,j}(X)$, and vanishes at $1$ when
 $j$ is odd.
 
-The roots of $Q_{n,j}$ are simple and positive. Kostov's Theorem 3 says that consecutive
-eigenpolynomials $Q_{n,j}$ and $Q_{n,j+1}$ strictly interlace, and his Theorem 4 compares
+The roots of $Q_{n,j}$ are simple and positive. Kostov's Theorem 3
+[@Kostov2010, Thm. 3] says that consecutive eigenpolynomials $Q_{n,j}$ and
+$Q_{n,j+1}$ strictly interlace, and his Theorem 4 [@Kostov2010, Thm. 4] compares
 the roots of $Q_{n,j}$ and $Q_{n,j+2}$, with the shared central root $1$ when $j$
 is odd. The proofs pass through the eigenproblem of a Bernstein-type operator, real zeros by
 a compactness argument, Descartes' rule of signs, and a Wronskian argument.
 
 ## References
 
-V. P. Kostov, “Interlacing properties and the Schur–Szegő composition,” *Functional
-Analysis and Other Mathematics* 3 (2010), 65–74; manuscript
-[arXiv:1504.02321](https://arxiv.org/abs/1504.02321).
+See [@Kostov2010].
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/Kurtz.lean b/RealRooted/Challenges/Kurtz.lean
index b53bcba0b..3f02e06d6 100644
--- a/RealRooted/Challenges/Kurtz.lean
+++ b/RealRooted/Challenges/Kurtz.lean
@@ -8,6 +8,9 @@ version = 1
 section = "theorems"
 slug = "kurtz"
 formalized = "2026-07-18"
+references = [
+  "Kurtz1992",
+]
 authors = ["Kurtz"]
 years = [1992]
 
@@ -38,14 +41,12 @@ records real-rootedness only.
 
 ## References
 
-D. C. Kurtz, [“A sufficient condition for all the roots of a polynomial to be
-real,”](https://doi.org/10.1080/00029890.1992.11995845) *American Mathematical
-Monthly* 99 (1992), 259–263. See also the
-[contextual account on symmetricfunctions.com](https://www.symmetricfunctions.com/realRooted.htm#kurtzTheorem).
+See [@Kurtz1992] and the
+[contextual account on symmetricfunctions.com](sf:kurtzTheorem).
 <!-- /realrooted-catalog-content -->
 
 Human statement:
-https://www.symmetricfunctions.com/realRooted.htm#kurtzTheorem
+sf:kurtzTheorem
 
 Original references: J. I. Hutchinson, "On a remarkable class of entire
 functions", Trans. Amer. Math. Soc. 25 (1923), 325--332, and D. C. Kurtz,
diff --git a/RealRooted/Challenges/LGV.lean b/RealRooted/Challenges/LGV.lean
index db824b3d4..f4e5d7295 100644
--- a/RealRooted/Challenges/LGV.lean
+++ b/RealRooted/Challenges/LGV.lean
@@ -11,6 +11,10 @@ version = 1
 section = "theorems"
 slug = "lindstrom-gessel-viennot"
 formalized = "2026-09-17"
+references = [
+  "Lindstrom1973",
+  "GesselViennot1985",
+]
 authors = ["Lindström", "Gessel", "Viennot"]
 years = [1973, 1985]
 
@@ -65,7 +69,7 @@ label = "Paths of length n and powers of the edge matrix"
 <!-- realrooted-catalog-content -->
 # Lindström–Gessel–Viennot and total positivity
 
-The [Lindström–Gessel–Viennot lemma](https://www.symmetricfunctions.com/lgv-lemma.htm#lindstromGesselViennotLemma)
+The [Lindström–Gessel–Viennot lemma](sf:lindstromGesselViennotLemma)
 expresses a path-matrix minor as a signed sum over vertex-disjoint path
 families. The formalization uses ordered cancellation certificates to show
 that the surviving terms have nonnegative weights.
@@ -87,11 +91,7 @@ Building on the path-cancellation proof in the LeanLGV library:
 
 ## References
 
-B. Lindström, “On the vector representations of induced matroids,” *Bull.
-London Math. Soc.* 5 (1973), 85–90.
-I. Gessel and G. Viennot, “Binomial determinants, paths, and hook length
-formulae,” *Advances in Mathematics* 58 (1985), 300–321.
-The kernel rows connect to the
+See [@Lindstrom1973; @GesselViennot1985]. The kernel rows connect to the
 [Brändén–Saud Leite page](/RealRooted/theorems/branden-leite/).
 <!-- /realrooted-catalog-content -->
 
diff --git a/RealRooted/Challenges/LaguerrePolarDerivative.lean b/RealRooted/Challenges/LaguerrePolarDerivative.lean
index 4a521d7ed..4adb2fc46 100644
--- a/RealRooted/Challenges/LaguerrePolarDerivative.lean
+++ b/RealRooted/Challenges/LaguerrePolarDerivative.lean
@@ -10,6 +10,9 @@ version = 1
 section = "theorems"
 slug = "laguerre-polar-derivative"
 formalized = "2026-07-01"
+references = [
+  "RahmanSchmeisser2002",
+]
 authors = ["Rahman", "Schmeisser"]
 years = [2002]
 
@@ -104,8 +107,7 @@ the derivative of a real-rooted polynomial is real-rooted by Rolle's theorem.
 
 ## References
 
-Q. I. Rahman and G. Schmeisser, *Analytic Theory of Polynomials*, Oxford University
-Press, 2002.
+See [@RahmanSchmeisser2002].
 <!-- /realrooted-catalog-content -->
 
 This module exposes Laguerre's theorem for closed disks, closed half-planes and the real
diff --git a/RealRooted/Challenges/LaguerrePolya.lean b/RealRooted/Challenges/LaguerrePolya.lean
index 19d7ff939..c84aea60d 100644
--- a/RealRooted/Challenges/LaguerrePolya.lean
+++ b/RealRooted/Challenges/LaguerrePolya.lean
@@ -16,6 +16,11 @@ version = 1
 section = "concepts"
 slug = "laguerre-polya-class"
 formalized = "2026-09-11"
+references = [
+  "SchurPolya1914",
+  "Levin1964",
+  "Hutchinson1923",
+]
 authors = ["Pólya", "Schur"]
 years = [1914]
 
@@ -214,12 +219,7 @@ $\varphi(D)\, g$. Real-rootedness passes to limits of polynomials of bounded deg
 
 ## References
 
-G. Pólya and J. Schur, “Über zwei Arten von Faktorenfolgen in der Theorie der
-algebraischen Gleichungen,” *J. Reine Angew. Math.* 144 (1914), 89–113;
-B. Ja. Levin, *Distribution of zeros of entire functions*, Chapter VIII;
-J. I. Hutchinson, “On a remarkable class of entire functions,” *Transactions
-of the American Mathematical Society* 25 (1923), 325–332,
-[doi:10.1090/s0002-9947-1923-1501248-1](https://doi.org/10.1090/s0002-9947-1923-1501248-1).
+See [@SchurPolya1914], [@Levin1964, Ch. VIII], and [@Hutchinson1923].
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade. The definition is in
diff --git a/RealRooted/Challenges/LaguerrePolynomials.lean b/RealRooted/Challenges/LaguerrePolynomials.lean
index a06d1e868..3bc6c99bd 100644
--- a/RealRooted/Challenges/LaguerrePolynomials.lean
+++ b/RealRooted/Challenges/LaguerrePolynomials.lean
@@ -10,6 +10,9 @@ version = 1
 section = "families"
 slug = "laguerre-polynomials"
 formalized = "2026-09-07"
+references = [
+  "Szego1939",
+]
 
 [[definitions]]
 name = "Polynomial.generalizedLaguerre"
@@ -74,9 +77,7 @@ $$
 
 ## References
 
-G. Szegő, [*Orthogonal Polynomials*](https://doi.org/10.1090/coll/023),
-American Mathematical Society Colloquium Publications 23 (1939), Chapter V.
-The general three-term theory is on the
+See [@Szego1939, Ch. V]. The general three-term theory is on the
 [Favard page](/RealRooted/theorems/favard/).
 <!-- /realrooted-catalog-content -->
 
diff --git a/RealRooted/Challenges/LeakeRyder.lean b/RealRooted/Challenges/LeakeRyder.lean
index 0e99477f8..0ad45b6e3 100644
--- a/RealRooted/Challenges/LeakeRyder.lean
+++ b/RealRooted/Challenges/LeakeRyder.lean
@@ -9,6 +9,9 @@ version = 1
 section = "theorems"
 slug = "leake-ryder"
 formalized = "2026-09-06"
+references = [
+  "LeakeRyder2019",
+]
 authors = ["Leake", "Ryder"]
 years = [2019]
 
@@ -53,8 +56,9 @@ label = "The matching polynomial is same-phase stable"
 # Same-phase stability for graph polynomials
 
 A multivariate polynomial is same-phase stable when every nonnegative
-one-dimensional specialization is real-rooted. Leake and Ryder proved that the
-multivariate independence polynomial is same-phase stable exactly for
+one-dimensional specialization is real-rooted. Leake and Ryder proved
+[@LeakeRyder2019] that the multivariate independence polynomial is same-phase
+stable exactly for
 claw-free graphs. Their result also gives same-phase stability of the
 edge-variable matching polynomial and recovers the Chudnovsky–Seymour theorem
 by setting every variable equal to the same univariate variable.
@@ -67,12 +71,7 @@ $1 + 4x + 3x^2 + x^3$, which has non-real roots.
 
 ## References
 
-J. D. Leake and N. R. Ryder, “Generalizations of the matching polynomial to
-the multivariate independence polynomial,” *Algebraic Combinatorics* 2 (2019),
-781–802. See the discussion of
-[same-phase stability][same-phase]
-on symmetricfunctions.com.
-
-[same-phase]: https://www.symmetricfunctions.com/stablePolynomials.htm#samePhaseStabilityDefinition
+See [@LeakeRyder2019] and the discussion of
+[same-phase stability](sf:samePhaseStabilityDefinition) on symmetricfunctions.com.
 <!-- /realrooted-catalog-content -->
 -/
diff --git a/RealRooted/Challenges/LeeYang.lean b/RealRooted/Challenges/LeeYang.lean
index cd41016ea..e3826c0b5 100644
--- a/RealRooted/Challenges/LeeYang.lean
+++ b/RealRooted/Challenges/LeeYang.lean
@@ -8,6 +8,11 @@ version = 1
 section = "theorems"
 slug = "lee-yang"
 formalized = "2026-10-07"
+references = [
+  "LeeYang1952",
+  "Asano1970",
+  "Ruelle2010",
+]
 authors = ["Lee", "Yang", "Asano", "Ruelle"]
 years = [1952, 1970]
 
@@ -44,12 +49,13 @@ P(z) = \sum_{S \subseteq V} \Bigl(\prod_{i \in S} \prod_{j \notin S} a_{ij}\Bigr
 $$
 has no zeros when every $|z_i| < 1$, and none when every $|z_i| > 1$.
 
-**Theorem (Lee–Yang).** All zeros of $P(z, z, \dotsc, z)$ lie on the unit circle $|z| = 1$.
+**Theorem (Lee–Yang).** [@LeeYang1952] All zeros of $P(z, z, \dotsc, z)$ lie on
+the unit circle $|z| = 1$.
 
 For the ferromagnetic Ising model, $a_{ij} = e^{-2\beta J_{ij}}$ with $J_{ij} \ge 0$, so the
 zeros of the partition function in the fugacity lie on the unit circle.
 
-The proof uses Asano contractions. The two-site polynomial $1 + c(z_1 + z_2) + z_1 z_2$
+The proof uses Asano contractions [@Asano1970]. The two-site polynomial $1 + c(z_1 + z_2) + z_1 z_2$
 with $|c| \le 1$ has no zeros in the open bidisc. If
 $\alpha + \beta u + \gamma v + \delta u v$ has no zeros there, then neither does
 $\alpha + \delta z$. Building $P$ edge by edge as a product of two-site factors in separate
@@ -58,13 +64,7 @@ exterior statement follows from the identity $P(z) = (\prod_i z_i)\, P(1/z)$.
 
 ## References
 
-T. D. Lee and C. N. Yang, [“Statistical theory of equations of state and phase transitions.
-II. Lattice gas and Ising model,”](https://doi.org/10.1103/PhysRev.87.410) *Physical Review*
-87 (1952), 410–419; T. Asano, [“Lee–Yang theorem and the Griffiths inequality for the
-anisotropic Heisenberg ferromagnet,”](https://doi.org/10.1103/PhysRevLett.24.1409) *Physical
-Review Letters* 24 (1970), 1409–1411; D. Ruelle, [“Characterization of Lee–Yang
-polynomials,”](https://doi.org/10.4007/annals.2010.171.589) *Annals of Mathematics* 171
-(2010), 589–603.
+See [@LeeYang1952; @Asano1970; @Ruelle2010].
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade. The proof lives in `RealRooted.LeeYang`.
diff --git a/RealRooted/Challenges/Liggett.lean b/RealRooted/Challenges/Liggett.lean
index b7027d673..a787bf9ab 100644
--- a/RealRooted/Challenges/Liggett.lean
+++ b/RealRooted/Challenges/Liggett.lean
@@ -8,6 +8,9 @@ version = 1
 section = "theorems"
 slug = "liggett-ultra-log-concavity"
 formalized = "2026-10-07"
+references = [
+  "Gurvits2009",
+]
 authors = ["Liggett"]
 years = [1997]
 
@@ -61,16 +64,13 @@ this ordering, by a $2 \times 2$ Cauchy–Binet argument as in Hoggar's theorem.
 A real-number inequality then combines the two induction hypotheses.
 
 This elementary argument is not Liggett's original proof. Other known proofs use
-mixed volumes (Gurvits), the Johnson scheme (Kahn–Neiman) or Lorentzian polynomials
+mixed volumes [@Gurvits2009], the Johnson scheme (Kahn–Neiman) or Lorentzian polynomials
 (Brändén–Huh).
 
 ## References
 
 T. M. Liggett, “Ultra logconcave sequences and negative dependence,” *Journal of
-Combinatorial Theory, Series A* 79 (1997), 315–325; L. Gurvits, [“A short proof,
-based on mixed volumes, of Liggett's theorem on the convolution of
-ultra-logconcave sequences,”](https://doi.org/10.37236/243) *Electronic Journal
-of Combinatorics* 16 (2009), #N5.
+Combinatorial Theory, Series A* 79 (1997), 315–325. See also [@Gurvits2009].
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in
diff --git a/RealRooted/Challenges/LiuOppositeSigns.lean b/RealRooted/Challenges/LiuOppositeSigns.lean
index 731156ab3..4aab32920 100644
--- a/RealRooted/Challenges/LiuOppositeSigns.lean
+++ b/RealRooted/Challenges/LiuOppositeSigns.lean
@@ -8,6 +8,9 @@ version = 1
 section = "theorems"
 slug = "liu-opposite-signs"
 formalized = "2026-10-02"
+references = [
+  "Liu2012CompatibleSequences",
+]
 authors = ["Liu"]
 years = [2012]
 
@@ -75,7 +78,7 @@ signs**, where the answer is a root-count condition.
 Write $n_p(x)$ for the number of roots of $p$ in $[x, \infty)$, counted with
 multiplicity.
 
-**Theorem (Liu, corrected).** Let $f$ and $g$ be nonconstant
+**Theorem (Liu, corrected).** [@Liu2012CompatibleSequences] Let $f$ and $g$ be nonconstant
 real-rooted polynomials with opposite leading signs. Then $f$ and $g$ are
 compatible if and only if one of the following holds:
 
@@ -99,9 +102,7 @@ matching passes back to the limit.
 
 ## References
 
-Lily L. Liu, [“Polynomials with real zeros and compatible
-sequences,”](https://doi.org/10.37236/2674) Electronic Journal of
-Combinatorics 19(3) (2012), #P33.
+See [@Liu2012CompatibleSequences].
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in
diff --git a/RealRooted/Challenges/MaWangLiuWang.lean b/RealRooted/Challenges/MaWangLiuWang.lean
index 0b0ea84d9..014096333 100644
--- a/RealRooted/Challenges/MaWangLiuWang.lean
+++ b/RealRooted/Challenges/MaWangLiuWang.lean
@@ -12,6 +12,10 @@ version = 1
 section = "theorems"
 slug = "ma-wang-liu-wang"
 formalized = "2026-10-06"
+references = [
+  "LiuWang2007",
+  "MaWang2008",
+]
 authors = ["Liu", "Wang", "Ma"]
 years = [2007, 2008]
 
@@ -70,7 +74,8 @@ the zeros interlace weakly and $F$ has the largest zero, so that
 $\deg F - \deg f \in \{0, 1\}$. Shared and repeated zeros are allowed. In
 particular, $f \ll F$ includes the statement that $F$ is real-rooted.
 
-**Theorem (Liu–Wang).** Let $f, g, a, b \in \mathbb{R}[x]$ and
+**Theorem (Liu–Wang).** [@LiuWang2007, Thm. 2.1] Let
+$f, g, a, b \in \mathbb{R}[x]$ and
 $F = a f + b g$. Suppose that $g \ll f$, that $g$ and $F$ have positive leading
 coefficients, and that $\deg f \leq \deg F \leq \deg f + 1$. If $b(r) \leq 0$
 at every real zero $r$ of $f$, then $f \ll F$.
@@ -87,7 +92,7 @@ $r$ of $f$, then $f \ll F$. Common zeros of $f$ and $F$ are allowed.
 Conversely, if $g \ll f$ and both have positive leading coefficients, then
 $g(r) f'(r) \geq 0$ at every zero $r$ of $f$.
 
-**Theorem (Ma–Wang).** Let $f, u, v \in \mathbb{R}[x]$ and
+**Theorem (Ma–Wang).** [@MaWang2008, Thm. 2] Let $f, u, v \in \mathbb{R}[x]$ and
 $F = u f + v f'$. Suppose that $f$ is real-rooted, that $f$ and $F$ have
 positive leading coefficients, and that $\deg f \leq \deg F \leq \deg f + 1$.
 If $v(r) \leq 0$ at every zero $r$ of $f$, then $f \ll F$.
@@ -129,14 +134,8 @@ stated here.
 
 ## References
 
-L. L. Liu and Y. Wang, [“A unified approach to polynomial sequences with only
-real zeros,”](https://doi.org/10.1016/j.aam.2006.02.003) *Advances in Applied
-Mathematics* 38 (2007), 542–560, Theorem 2.1.
-S.-M. Ma and Y. Wang, [“q-Eulerian polynomials and polynomials with only real
-zeros,”](https://doi.org/10.37236/741) *Electronic Journal of Combinatorics*
-15 (2008), #R17, Theorem 2.
-See also the
-[Ma–Wang and Liu–Wang criteria on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedInterlacing.htm#maWangDerivativeCriterion).
+See [@LiuWang2007, Thm. 2.1] and [@MaWang2008, Thm. 2]. See also the
+[Ma–Wang and Liu–Wang criteria on symmetricfunctions.com](sf:maWangDerivativeCriterion).
 <!-- /realrooted-catalog-content -->
 
 The general Liu–Wang theorem lives in `RealRooted.LiuWang.General`, the
diff --git a/RealRooted/Challenges/MatrixInterlacing.lean b/RealRooted/Challenges/MatrixInterlacing.lean
index fdbe9f2e4..df6d2882f 100644
--- a/RealRooted/Challenges/MatrixInterlacing.lean
+++ b/RealRooted/Challenges/MatrixInterlacing.lean
@@ -9,6 +9,10 @@ version = 1
 section = "theorems"
 slug = "matrix-interlacing"
 formalized = "2026-10-06"
+references = [
+  "Fisk2006x",
+  "Branden2015",
+]
 authors = ["Fisk", "Brändén"]
 years = [2006, 2015]
 
@@ -97,15 +101,13 @@ zero-aware forward forms.
 
 ## References
 
-S. Fisk, *Polynomials, roots, and interlacing*, 2006, Chapter 3, develops
-matrices preserving interlacing. P. Brändén gives the exact nonnegative
-polynomial characterization in “Unimodality, log-concavity, real-rootedness
-and beyond,” *Handbook of Enumerative Combinatorics*, 2015, Theorem 7.8.5. See the
-[matrix criterion on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedInterlacing.htm#matrixPreservesInterlacingSequences).
+Matrices preserving interlacing are treated in [@Fisk2006x, Ch. 3]. The exact
+nonnegative polynomial characterization is [@Branden2015, Thm. 7.8.5]. See the
+[matrix criterion on symmetricfunctions.com](sf:matrixPreservesInterlacingSequences).
 <!-- /realrooted-catalog-content -->
 
 Human statement:
-https://www.symmetricfunctions.com/realRootedInterlacing.htm#matrixPreservesInterlacingSequences
+sf:matrixPreservesInterlacingSequences
 
 References: S. Fisk, *Polynomials, roots, and interlacing* (2006), Chapter 3;
 P. Branden, *Unimodality, log-concavity, real-rootedness and beyond*,
diff --git a/RealRooted/Challenges/MonomialChainOperator.lean b/RealRooted/Challenges/MonomialChainOperator.lean
index afd743822..088676746 100644
--- a/RealRooted/Challenges/MonomialChainOperator.lean
+++ b/RealRooted/Challenges/MonomialChainOperator.lean
@@ -11,6 +11,9 @@ version = 1
 section = "theorems"
 slug = "monomial-chain-operator"
 formalized = "2026-05-12"
+references = [
+  "Branden2011b",
+]
 
 [[theorems]]
 name = "RealRooted.Challenges.MonomialChainOperator.preservesInterlacing"
@@ -67,9 +70,7 @@ oriented interlacing pair.
 ## References
 
 The decomposition step uses the Krein expansion for interlacing polynomials;
-the operator perspective is related to P. Brändén, [“Iterated sequences and
-the geometry of zeros,”](https://doi.org/10.1515/crelle.2011.063) *Journal für
-die reine und angewandte Mathematik* 658 (2011), 115–131.
+the operator perspective is related to [@Branden2011b].
 <!-- /realrooted-catalog-content -->
 
 This module exposes the fully checked monomial-chain upgrade.
diff --git a/RealRooted/Challenges/MonotoneColumnPermanent.lean b/RealRooted/Challenges/MonotoneColumnPermanent.lean
index 0f83a1bdd..73459ce9a 100644
--- a/RealRooted/Challenges/MonotoneColumnPermanent.lean
+++ b/RealRooted/Challenges/MonotoneColumnPermanent.lean
@@ -8,6 +8,10 @@ version = 1
 section = "theorems"
 slug = "monotone-column-permanent"
 formalized = "2026-10-13"
+references = [
+  "HaglundOnoWagner1999",
+  "BrandenHaglundVisontaiWagner2011",
+]
 authors = ["Haglund", "Ono", "Wagner", "Brändén", "Visontai"]
 years = [1999, 2011]
 
@@ -95,7 +99,7 @@ half-plane.
 Reversing the rows preserves the permanent and changes decreasing columns into
 increasing columns. Thus
 `columnPermanent_mvRealStable_of_increasingColumns` gives the convention used
-in the original Haglund–Ono–Wagner conjecture.
+in the original Haglund–Ono–Wagner conjecture [@HaglundOnoWagner1999].
 
 Setting every $z_j$ equal to one variable $z$ gives
 $$
@@ -118,18 +122,7 @@ fractional substitution yield the multivariate theorem.
 
 ## References
 
-J. Haglund, K. Ono, and D. G. Wagner,
-[“Theorems and conjectures involving rook polynomials with only real
-zeros,”](https://doi.org/10.1007/978-1-4613-0305-3_13) in *Topics in Number
-Theory*, Mathematics and Its Applications 467 (1999), 207–221,
-[doi:10.1007/978-1-4613-0305-3_13](https://doi.org/10.1007/978-1-4613-0305-3_13).
-
-P. Brändén, J. Haglund, M. Visontai, and D. G. Wagner,
-[“Proof of the monotone column permanent
-conjecture,”](https://arxiv.org/abs/1010.2565) in *Notions of Positivity and
-the Geometry of Polynomials*, Trends in Mathematics (2011), 63–78,
-[doi:10.1007/978-3-0348-0142-3_5](https://doi.org/10.1007/978-3-0348-0142-3_5),
-[arXiv:1010.2565](https://arxiv.org/abs/1010.2565).
+See [@HaglundOnoWagner1999; @BrandenHaglundVisontaiWagner2011].
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/MorrisFlow.lean b/RealRooted/Challenges/MorrisFlow.lean
index 9a1148c94..9668769e7 100644
--- a/RealRooted/Challenges/MorrisFlow.lean
+++ b/RealRooted/Challenges/MorrisFlow.lean
@@ -10,6 +10,9 @@ version = 1
 section = "theorems"
 slug = "morris-flow"
 formalized = "2026-10-16"
+references = [
+  "LiuZhang2026bx",
+]
 authors = ["Liu", "Zhang"]
 years = [2026]
 
@@ -85,8 +88,8 @@ headline = true
 <!-- realrooted-catalog-content -->
 # The Morris flow and its stable eigenfunctions
 
-Following Liu and Zhang, we work with symmetric polynomials in $x_1, \dotsc, x_q$ and the
-differential operators
+Following Liu and Zhang [@LiuZhang2026bx], we work with symmetric polynomials in
+$x_1, \dotsc, x_q$ and the differential operators
 $$
 A_q = \sum_{i=1}^{q} x_i^2 \partial_i^2, \qquad
 B_q = 2 \sum_{1 \leq i < j \leq q}
@@ -100,7 +103,8 @@ nonzero (see [real stability](/RealRooted/theorems/real-stability/)). We write
 $\operatorname{Sym}_{q,\leq N}$ for the finite-dimensional real space of symmetric polynomials
 in $q$ variables of total degree at most $N$.
 
-**Theorem** (Liu–Zhang, Theorem 4.9). Let $t \geq 0$ and $N \in \mathbb N$. If
+**Theorem** (Liu–Zhang, Theorem 4.9). [@LiuZhang2026bx, Thm. 4.9]
+Let $t \geq 0$ and $N \in \mathbb N$. If
 $P \in \operatorname{Sym}_{q,\leq N}$ is real stable, then $e^{t D_q} P$ is real stable.
 
 In the Lean statement, $e^{tD_q}$ is the exponential of the restriction of $D_q$ to
@@ -123,7 +127,8 @@ $$
 $N$. A coefficientwise limit of this flow as $t = n \to \infty$ through the natural numbers,
 which is nonzero and of total degree at most $N$, is real stable.
 
-**Theorem** (Liu–Zhang, Theorem 4.12). Let $\lambda$ be a partition of $N$ with at most $q$
+**Theorem** (Liu–Zhang, Theorem 4.12). [@LiuZhang2026bx, Thm. 4.12]
+Let $\lambda$ be a partition of $N$ with at most $q$
 parts such that $\lambda_i - \lambda_{i+1} \leq 1$ for $1 \leq i < q$. Let $V_\lambda$ be the
 span of the monomial symmetric polynomials $m_\mu$ for partitions $\mu \preceq \lambda$ in
 dominance order with at most $q$ parts.
@@ -154,9 +159,8 @@ to every normalized eigenfunction.
 
 ## References
 
-F. Liu and Z. Zhang, [*Real-rootedness and gamma-positivity for a variation of the Morris
-constant term*](https://arxiv.org/abs/2609.15201), arXiv:2609.15201 (2026). Theorems 4.9 and
-4.12 are Theorems 1.2 and 1.3 of the paper, and Lemma 4.4 is the real stability of $E_\lambda$.
+See [@LiuZhang2026bx]. Theorems 4.9 and 4.12 here are Theorems 1.2 and 1.3
+of that paper, and Lemma 4.4 is its real-stability result for $E_\lambda$.
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade. The operators and the eigenvalue are in
diff --git a/RealRooted/Challenges/MultiplierSequence.lean b/RealRooted/Challenges/MultiplierSequence.lean
index ca961bbf8..7624d36d8 100644
--- a/RealRooted/Challenges/MultiplierSequence.lean
+++ b/RealRooted/Challenges/MultiplierSequence.lean
@@ -18,6 +18,9 @@ version = 1
 section = "concepts"
 slug = "multiplier-sequences"
 formalized = "2026-09-07"
+references = [
+  "SchurPolya1914",
+]
 authors = ["Pólya", "Schur"]
 years = [1914]
 
@@ -221,9 +224,8 @@ each approximating polynomial, and the Taylor coefficients converge.
 
 ## References
 
-G. Pólya and J. Schur, “Über zwei Arten von Faktorenfolgen in der Theorie der
-algebraischen Gleichungen,” *J. Reine Angew. Math.* 144 (1914), 89–113.
-The finite Pólya–Schur theorem through the Schur–Szegő composition is on the
+See [@SchurPolya1914]. The finite Pólya–Schur theorem through the
+Schur–Szegő composition is on the
 [Hadamard products page](/RealRooted/theorems/hadamard-products/).
 <!-- /realrooted-catalog-content -->
 
diff --git a/RealRooted/Challenges/MultisetEulerianNarayana.lean b/RealRooted/Challenges/MultisetEulerianNarayana.lean
index 1fa78ad4b..1768714e8 100644
--- a/RealRooted/Challenges/MultisetEulerianNarayana.lean
+++ b/RealRooted/Challenges/MultisetEulerianNarayana.lean
@@ -11,6 +11,10 @@ version = 1
 section = "families"
 slug = "multiset-eulerian-narayana"
 formalized = "2026-10-07"
+references = [
+  "ZhangZhao2026x",
+  "LinMaMaZhou2021",
+]
 authors = ["Lin", "Ma", "Zhou", "Zhang", "Zhao"]
 years = [2021, 2026]
 
@@ -72,7 +76,7 @@ label = "Zhang–Zhao: the closed transition matrices are totally nonnegative"
 Let $M = \{1^{p_1}, \dotsc, m^{p_m}\}$ be a multiset of size $N$. A weakly
 increasing tree on $M$ is a plane tree on $M \cup \{0\}$, rooted at $0$, whose
 labels weakly increase along every root-to-leaf path and from left to right
-among siblings. Lin, Ma, Ma and Zhou studied the normalized leaf enumerator
+among siblings. Lin, Ma, Ma and Zhou [@LinMaMaZhou2021] studied the normalized leaf enumerator
 $P_M(t)$ of these trees. It interpolates between the Eulerian polynomials
 ($p_i = 1$) and the Narayana polynomials ($m = 1$). They conjectured that it is
 real-rooted.
@@ -89,18 +93,20 @@ $$
 Here this factorization defines $P_\alpha$. The tree model and the
 Lagrange–Bürmann argument behind the factorization are not formalized.
 
-**Theorem** (Zhang–Zhao). For every composition $\alpha$ of $N \ge 1$, the
+**Theorem** (Zhang–Zhao). [@ZhangZhao2026x] For every composition $\alpha$ of
+$N \ge 1$, the
 polynomial $P_\alpha$ has degree $N - 1$, nonnegative coefficients and only
 simple negative zeros. Consecutive partial products of the factorization strictly
 interlace, and $P_\alpha$ is palindromic.
 
-**Theorem** (Zhang–Zhao, Theorem 1.3). The matrices $K^{(p,r)}$ that send the
+**Theorem** (Zhang–Zhao, Theorem 1.3). [@ZhangZhao2026x, Thm. 1.3] The
+matrices $K^{(p,r)}$ that send the
 coefficients of an input polynomial to those of its image under $r$ consecutive steps
 have an explicit binomial closed form. They are products of positive bidiagonal
 matrices, hence totally nonnegative.
 
 The conjecture itself also needs the identification of $P_\alpha$ with the leaf
-enumerator $P_M$ (Zhang–Zhao, Proposition 2.9). That identification is cited
+enumerator $P_M$ [@ZhangZhao2026x, Prop. 2.9]. That identification is cited
 from the paper, not formalized.
 
 ## Proof idea
@@ -113,12 +119,9 @@ $a_d(\alpha)$ are positive.
 
 ## References
 
-Z. Lin, J. Ma, S.-M. Ma and Y. Zhou, [“Weakly increasing trees on a multiset,”](https://doi.org/10.1016/j.aam.2021.102206)
-*Advances in Applied Mathematics* 129 (2021), 102206; P. B. Zhang and T. Zhao,
-[“Zeros and interlacing for multiset Eulerian–Narayana polynomials,”](https://arxiv.org/abs/2610.00966)
-arXiv:2610.00966 (2026).
-
-The real-rootedness theorem was formalized in RealRooted issue #1076.
+For the tree model, see [@LinMaMaZhou2021]. For the interlacing result, see
+[@ZhangZhao2026x]. The real-rootedness theorem was formalized in RealRooted
+issue #1076.
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in
diff --git a/RealRooted/Challenges/MultivariateObreschkoff.lean b/RealRooted/Challenges/MultivariateObreschkoff.lean
index f6c395732..766b8e70e 100644
--- a/RealRooted/Challenges/MultivariateObreschkoff.lean
+++ b/RealRooted/Challenges/MultivariateObreschkoff.lean
@@ -10,6 +10,11 @@ version = 1
 section = "theorems"
 slug = "multivariate-obreschkoff"
 formalized = "2026-10-13"
+references = [
+  "BorceaBranden2009I",
+  "BorceaBranden2010",
+  "Obreschkoff1963",
+]
 authors = ["Obreschkoff", "Borcea", "Brändén"]
 years = [1963, 2009, 2010]
 
@@ -57,7 +62,8 @@ polynomial; in particular it is nonzero.
 For $F, G \in \mathbb R[z_i : i \in \sigma]$ we say that $F$ and $G$ are in
 **proper position**, written $F \ll G$, if the complex polynomial $G + iF$ is stable.
 
-**Theorem** (Hermite–Kakeya–Obreschkoff, converse). Let $\sigma$ be any index type and
+**Theorem** (Hermite–Kakeya–Obreschkoff, converse). [@Obreschkoff1963] Let
+$\sigma$ be any index type and
 $F, G \in \mathbb R[z_i : i \in \sigma]$. Suppose that for all real $\alpha, \beta$ the
 polynomial $\alpha F + \beta G$ is zero or real stable. Then $F = G = 0$, or
 $F \ll G$, or $G \ll F$.
@@ -94,17 +100,8 @@ $G + iF$ stable and a negative sign makes $F + iG$ stable.
 
 ## References
 
-J. Borcea and P. Brändén,
-[“The Lee–Yang and Pólya–Schur programs. I. Linear operators preserving
-stability,”](https://arxiv.org/abs/0809.0401) *Invent. Math.* 177 (2009), 541–569,
-[doi:10.1007/s00222-009-0189-3](https://doi.org/10.1007/s00222-009-0189-3),
-Lemma 1.8 and Theorem 1.9 (arXiv:0809.0401). The theorem was first proved in
-J. Borcea and P. Brändén,
-[“Multivariate Pólya–Schur classification problems in the Weyl
-algebra,”](https://arxiv.org/abs/math/0606360) *Proc. London Math. Soc.* (3) 101 (2010),
-73–104, [doi:10.1112/plms/pdp049](https://doi.org/10.1112/plms/pdp049). For the univariate
-theorem see N. Obreschkoff, *Verteilung und Berechnung der Nullstellen reeller Polynome*,
-VEB Deutscher Verlag der Wissenschaften, 1963.
+See [@BorceaBranden2009I, Lemma 1.8 and Thm. 1.9],
+[@BorceaBranden2010], and [@Obreschkoff1963].
 <!-- /realrooted-catalog-content -->
 
 This module exposes the two directions of the multivariate Hermite–Kakeya–Obreschkoff
diff --git a/RealRooted/Challenges/Narayana.lean b/RealRooted/Challenges/Narayana.lean
index fb4433c3d..e9540e7f9 100644
--- a/RealRooted/Challenges/Narayana.lean
+++ b/RealRooted/Challenges/Narayana.lean
@@ -14,6 +14,12 @@ version = 1
 section = "families"
 slug = "narayana"
 formalized = "2026-07-18"
+references = [
+  "MaoWang2026x",
+  "Brenti1989",
+  "DominiciJohnstonJordaan2013x",
+  "Zeng2026x",
+]
 authors = ["Mao", "Wang", "Dominici", "Johnston", "Jordaan", "Brenti", "Zeng"]
 years = [1989, 2013, 2026]
 
@@ -170,7 +176,7 @@ $$
 for the falling factorials and the generalized rising factorials, and $T_n(x)$
 for the Touchard polynomials, $T_0 = 1$ and $T_{n+1} = x T_n + x T_n'$.
 
-**Theorem (Brenti).** If
+**Theorem (Brenti).** [@Brenti1989, Thm. 2.4.2] If
 $\sum_k a_k \langle x\rangle_k$ has only nonpositive zeros, then so does
 $\sum_k a_k x^k$.
 
@@ -190,7 +196,8 @@ $(ax + d)^n$ with $a > 0$ and $d \geq 0$, $T_n(x)$, $(x \mid \mu)_n$ with
 $\mu > 0$, and $N_{n,m}(x)$; row $n$ of the coefficient matrix of a basis
 $(B_n)$ lists the coefficients of $B_n$.
 
-**Theorem (Mao–Wang).** Let $M$ be lower-triangular with Pólya-frequency
+**Theorem (Mao–Wang).** [@MaoWang2026x, Thm. 1.3] Let $M$ be lower-triangular
+with Pólya-frequency
 row-generating polynomials, and let $B_1, \dotsc, B_r$ be admissible factors.
 Then $M B_1 \dotsm B_r$ has Pólya-frequency row-generating polynomials. In
 particular this holds for $M B^r$ with $B$ admissible and $r \geq 0$.
@@ -223,24 +230,14 @@ real-rootedness for $\theta \ge 0$ through Jacobi polynomials.
 
 ## References
 
-The coefficient normalization is from Jianxi Mao and Lijie Wang, [“The
-Narayana transformation,”](https://arxiv.org/abs/2607.01572) arXiv:2607.01572
-(2026).  The root-location input is D. Dominici, S. J. Johnston,
-and K. Jordaan, [“Real zeros of 2F1 hypergeometric
-polynomials,”](https://arxiv.org/abs/1301.4771) *Journal of Computational and
-Applied Mathematics* 247 (2013), 152–161.  The falling-factorial theorem is
-F. Brenti, [“Unimodal, log-concave and Pólya frequency sequences in
-combinatorics,”](https://doi.org/10.1090/memo/0413) *Memoirs of the American
-Mathematical Society* 81 (1989), no. 413, Theorem 2.4.2; Mao and Wang
-use it as Lemma 3.2, and the matrix-product theorem is their Theorem 1.3.
-The continuous interpolation is from J. Zeng, [“Continuous Narayana interpolations:
-Hankel total positivity and log-convexity preservation,”](https://arxiv.org/abs/2610.19548)
-arXiv:2610.19548 (2026), (1.1)–(1.3) and Corollary 3.5.
-See also the
-[Narayana real-rootedness examples on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedCatalan.htm#ex:narayanaSturm),
-[Brenti's falling-factorial theorem](https://www.symmetricfunctions.com/realRooted.htm#thm:brentiFallingFactorialBasis),
-and the
-[Mao–Wang row-generating functions](https://www.symmetricfunctions.com/realRooted.htm#rowGeneratingFunction).
+The coefficient normalization is from [@MaoWang2026x], and the root-location input
+is [@DominiciJohnstonJordaan2013x]. The falling-factorial theorem is
+[@Brenti1989, Thm. 2.4.2]; Mao and Wang use it as Lemma 3.2, and the
+matrix-product theorem is [@MaoWang2026x, Thm. 1.3]. The continuous interpolation
+is [@Zeng2026x, (1.1)–(1.3) and Cor. 3.5]. See also the
+[Narayana real-rootedness examples on symmetricfunctions.com](sf:ex:narayanaSturm),
+[Brenti's falling-factorial theorem](sf:thm:brentiFallingFactorialBasis), and the
+[Mao–Wang row-generating functions](sf:rowGeneratingFunction).
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/NewtonDefectRoots.lean b/RealRooted/Challenges/NewtonDefectRoots.lean
index e208767e5..fbdef8b98 100644
--- a/RealRooted/Challenges/NewtonDefectRoots.lean
+++ b/RealRooted/Challenges/NewtonDefectRoots.lean
@@ -8,6 +8,10 @@ version = 1
 section = "theorems"
 slug = "local-newton-defect-roots"
 formalized = "2026-10-11"
+references = [
+  "KarpenkoVishnyakova2017",
+  "AkianGaubertSharify2017",
+]
 
 [[theorems]]
 name = "RealRooted.NewtonDefectRoots.esym_sq_sub_sq_esym_le"
@@ -81,14 +85,7 @@ ratio; no bound on the ambient degrees is assumed.
 
 ## References
 
-I. Karpenko and A. Vishnyakova, “On sufficient conditions for a polynomial to
-be sign-independently hyperbolic or to have real separated zeros,” *Mathematical
-Inequalities & Applications* 20 (2017),
-[doi:10.7153/mia-20-18](https://doi.org/10.7153/mia-20-18).
-M. Akian, S. Gaubert, and M. Sharify, “Log-majorization of the moduli of the
-eigenvalues of a matrix polynomial by tropical roots,” *Linear Algebra and its
-Applications*, [doi:10.1016/j.laa.2016.11.004](https://doi.org/10.1016/j.laa.2016.11.004),
-[arXiv:1304.2967](https://arxiv.org/abs/1304.2967).
+See [@KarpenkoVishnyakova2017; @AkianGaubertSharify2017].
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/Nijenhuis.lean b/RealRooted/Challenges/Nijenhuis.lean
index 2d8000e08..def144d01 100644
--- a/RealRooted/Challenges/Nijenhuis.lean
+++ b/RealRooted/Challenges/Nijenhuis.lean
@@ -8,6 +8,9 @@ version = 1
 section = "theorems"
 slug = "nijenhuis-rook-polynomials"
 formalized = "2026-09-26"
+references = [
+  "Nijenhuis1976",
+]
 authors = ["Nijenhuis"]
 years = [1976]
 
@@ -53,10 +56,8 @@ with nonpositive roots; Nijenhuis’s signed normalization has nonnegative roots
 
 ## References
 
-A. Nijenhuis, [“On permanents and the zeros of rook polynomials,”](https://doi.org/10.1016/0097-3165(76)90068-6)
-*Journal of Combinatorial Theory, Series A* 21 (1976), 240–244. See also the
-[rook-polynomial overview on
-symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedGraphs.htm#rookPolynomials).
+See [@Nijenhuis1976] and the
+[rook-polynomial overview on symmetricfunctions.com](sf:rookPolynomials).
 <!-- /realrooted-catalog-content -->
 
 This challenge entry uses ordinary nonattacking rook placements. It is
diff --git a/RealRooted/Challenges/Obreschkoff.lean b/RealRooted/Challenges/Obreschkoff.lean
index 57ecb1a83..162753f68 100644
--- a/RealRooted/Challenges/Obreschkoff.lean
+++ b/RealRooted/Challenges/Obreschkoff.lean
@@ -10,6 +10,10 @@ version = 1
 section = "theorems"
 slug = "obreschkoff"
 formalized = "2026-07-10"
+references = [
+  "Dedieu1992",
+  "Obreschkoff1963",
+]
 authors = ["Obreschkoff", "Dedieu"]
 years = [1963, 1992]
 
@@ -45,7 +49,8 @@ label = "Interlacing as stability of the bivariate pencil f(z) + w g(z)"
 <!-- realrooted-catalog-content -->
 # Obreschkoff’s theorem
 
-Two interlacing polynomials generate a real-rooted pencil: every real linear
+Obreschkoff's theorem [@Obreschkoff1963; @Dedieu1992] says that two interlacing
+polynomials generate a real-rooted pencil: every real linear
 combination $\alpha f + \beta g$ splits over $\mathbb R$. Conversely, if
 every real linear combination of $f$ and $g$ splits, then $f$ and $g$
 interlace in one of the two orientations, where either polynomial may be zero.
@@ -64,15 +69,12 @@ $w = i$ and uses the Hermite–Biehler theorem.
 
 ## References
 
-N. Obreschkoff, *Verteilung und Berechnung der Nullstellen reeller
-Polynome*, VEB Deutscher Verlag der Wissenschaften, 1963; J.-P. Dedieu,
-“Obreschkoff’s theorem revisited,” *Journal of Pure and Applied Algebra* 81
-(1992), 269–278.  See the
-[contextual statement on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedInterlacing.htm#obreschkoffDedieu).
+See [@Obreschkoff1963; @Dedieu1992] and the
+[contextual statement on symmetricfunctions.com](sf:obreschkoffDedieu).
 <!-- /realrooted-catalog-content -->
 
 Human statement:
-https://www.symmetricfunctions.com/realRootedInterlacing.htm#obreschkoffDedieu
+sf:obreschkoffDedieu
 
 Original references include N. Obreschkoff, *Verteilung und Berechnung der
 Nullstellen reeller Polynome* (1963), and J.-P. Dedieu, "Obreschkoff's theorem
diff --git a/RealRooted/Challenges/OperatorPreservers.lean b/RealRooted/Challenges/OperatorPreservers.lean
index e24040d58..1451900b8 100644
--- a/RealRooted/Challenges/OperatorPreservers.lean
+++ b/RealRooted/Challenges/OperatorPreservers.lean
@@ -16,6 +16,10 @@ version = 1
 section = "theorems"
 slug = "operator-preservers"
 formalized = "2026-10-05"
+references = [
+  "Branden2004operators",
+  "Branden2006",
+]
 authors = ["Brändén"]
 years = [2004]
 
@@ -119,18 +123,13 @@ $ab > 0$.
 
 ## References
 
-P. Brändén, [“On operators on polynomials preserving real-rootedness and the
-Neggers–Stanley conjecture,”](https://arxiv.org/abs/math/0303147) *Journal of
-Algebraic Combinatorics* 20 (2004), 119–130 (Theorem 8 in the arXiv version).
-The strict-to-weak transfer follows P. Brändén, [“On linear transformations
-preserving the Pólya frequency property,”](https://arxiv.org/abs/math/0403364)
-*Transactions of the American Mathematical Society* 358 (2006), Corollary 3.5.
-See also the
-[operator-preserver overview on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedInterlacing.htm#operatorPreservesInterlacing).
+The finite-degree criterion is [@Branden2004operators, Thm. 8 in the arXiv
+version]. The strict-to-weak transfer is [@Branden2006, Cor. 3.5]. See also the
+[operator-preserver overview on symmetricfunctions.com](sf:operatorPreservesInterlacing).
 <!-- /realrooted-catalog-content -->
 
 Human statement:
-https://www.symmetricfunctions.com/realRootedInterlacing.htm#operatorPreservesInterlacing
+sf:operatorPreservesInterlacing
 
 Catalog reference: P. Brändén, J. Algebraic Combin. 20 (2004), 119–130,
 arXiv:math/0303147; the site cites it as Theorem 9.
diff --git a/RealRooted/Challenges/ParkingFunctions.lean b/RealRooted/Challenges/ParkingFunctions.lean
index 64d066d7e..09ac56693 100644
--- a/RealRooted/Challenges/ParkingFunctions.lean
+++ b/RealRooted/Challenges/ParkingFunctions.lean
@@ -10,6 +10,9 @@ version = 1
 section = "families"
 slug = "parking-functions"
 formalized = "2026-09-13"
+references = [
+  "DiaconisHicks2017",
+]
 
 [[definitions]]
 name = "ParkingFunction.IsParkingFunction"
@@ -64,8 +67,8 @@ headline = true
 <!-- realrooted-catalog-content -->
 # Parking functions
 
-A [parking function](https://www.symmetricfunctions.com/parking-functions.htm#parkingFunctionDefinition)
-of length $n$ is a word $w$ with letters in
+A [parking function](sf:parkingFunctionDefinition)
+of length $n$ [@DiaconisHicks2017] is a word $w$ with letters in
 $\{0, \dotsc, n-1\}$ such that, for every $k \leq n$, at least $k$ of its letters are
 smaller than $k$. The following generating polynomials over parking functions
 of length $n$ are real-rooted:
@@ -80,9 +83,7 @@ of length $n$ are real-rooted:
 
 ## References
 
-P. Diaconis and A. Hicks, “Probabilizing parking functions,” *Adv. in Appl.
-Math.* 89 (2017), 125–155.
-The Chow polynomials appear on the
+See [@DiaconisHicks2017]. The Chow polynomials appear on the
 [Brändén–Vecchi page](/RealRooted/theorems/branden-vecchi/).
 <!-- /realrooted-catalog-content -->
 
diff --git a/RealRooted/Challenges/PathPowerCliques.lean b/RealRooted/Challenges/PathPowerCliques.lean
index 1e35fb471..0a1e00a58 100644
--- a/RealRooted/Challenges/PathPowerCliques.lean
+++ b/RealRooted/Challenges/PathPowerCliques.lean
@@ -9,6 +9,9 @@ version = 1
 section = "theorems"
 slug = "path-power-cliques-and-rows"
 formalized = "2026-10-13"
+references = [
+  "ChudnovskySeymour2007",
+]
 
 [[definitions]]
 name = "RealRooted.pathPowerCliquePolynomial"
@@ -158,13 +161,11 @@ interlace.
 
 ## References
 
-M. Chudnovsky and P. Seymour, “The roots of the independence polynomial of a clawfree graph,”
-*Journal of Combinatorial Theory, Series B* 97 (2007), 350–357 (real-rootedness for claw-free
-graphs).  The row families are OEIS
-[A193636](https://oeis.org/A193636), [A109955](https://oeis.org/A109955),
+Real-rootedness follows from [@ChudnovskySeymour2007]. The row families are
+OEIS [A193636](https://oeis.org/A193636), [A109955](https://oeis.org/A109955),
 [A127893](https://oeis.org/A127893), [A206294](https://oeis.org/A206294),
 [A109960](https://oeis.org/A109960), [A116088](https://oeis.org/A116088),
-[A116089](https://oeis.org/A116089) and [A277627](https://oeis.org/A277627).
+[A116089](https://oeis.org/A116089), and [A277627](https://oeis.org/A277627).
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/PeakPolynomials.lean b/RealRooted/Challenges/PeakPolynomials.lean
index 35cbce1b3..1ea2b5827 100644
--- a/RealRooted/Challenges/PeakPolynomials.lean
+++ b/RealRooted/Challenges/PeakPolynomials.lean
@@ -145,7 +145,7 @@ polynomials depends on an open conjecture and is not part of this page.
 P. Alexandersson, A. Jal and M. Quemener, “Rook-Eulerian polynomials and
 permutation ideals,” manuscript, Theorem `thm:peak_rr_ferrers` and
 Lemma `lem:t_minus_1`. For peaks of permutations, see the
-[entry on symmetricfunctions.com](https://www.symmetricfunctions.com/gammaPositivity.htm#permutationPeak).
+[entry on symmetricfunctions.com](sf:permutationPeak).
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade. The proofs live in
diff --git a/RealRooted/Challenges/PerronFrobenius.lean b/RealRooted/Challenges/PerronFrobenius.lean
index 12d01c3a8..3284ddd5b 100644
--- a/RealRooted/Challenges/PerronFrobenius.lean
+++ b/RealRooted/Challenges/PerronFrobenius.lean
@@ -11,6 +11,9 @@ version = 1
 section = "theorems"
 slug = "perron-frobenius"
 formalized = "2026-08-21"
+references = [
+  "Perron1907",
+]
 authors = ["Perron", "Frobenius"]
 years = [1907, 1912]
 
@@ -58,7 +61,7 @@ $$
 r(A) = \sup_{v \geq 0,\ v \neq 0}\ \min_{i\,:\,v_i > 0} \frac{(Av)_i}{v_i}.
 $$
 
-**Theorem (Perron–Frobenius).**
+**Theorem (Perron–Frobenius).** [@Perron1907]
 
 - Every nonnegative matrix has $r(A)$ as an eigenvalue, with a nonnegative
   eigenvector.
@@ -76,11 +79,9 @@ one through $1 + A$.
 
 ## References
 
-O. Perron, [“Zur Theorie der Matrices,”](https://doi.org/10.1007/BF01449896)
-*Mathematische Annalen* 64 (1907), 248–263.
-
-G. Frobenius, “Über Matrizen aus nicht negativen Elementen,” Sitzungsberichte
-der Königlich Preussischen Akademie der Wissenschaften (1912), 456–477.
+See [@Perron1907]. G. Frobenius, “Über Matrizen aus nicht negativen Elementen,”
+*Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften* (1912),
+456–477.
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in
diff --git a/RealRooted/Challenges/ProperPositionPF.lean b/RealRooted/Challenges/ProperPositionPF.lean
index f92407b89..4af6550dd 100644
--- a/RealRooted/Challenges/ProperPositionPF.lean
+++ b/RealRooted/Challenges/ProperPositionPF.lean
@@ -8,6 +8,11 @@ version = 1
 section = "theorems"
 slug = "pf-multiplier-proper-position"
 formalized = "2026-10-11"
+references = [
+  "BorceaBranden2009I",
+  "BorceaBranden2010",
+  "Branden2015",
+]
 
 [[theorems]]
 name = "RealRooted.IsFinitePFMultiplierSequence.interl_diagonalOperator"
@@ -75,16 +80,7 @@ nonzero splitness properties.
 
 ## References
 
-J. Borcea and P. Brändén, [“The Lee–Yang and Pólya–Schur programs. I. Linear
-operators preserving stability,”](https://doi.org/10.1007/s00222-009-0189-3)
-*Inventiones Mathematicae* 177 (2009), 541–569,
-[arXiv:0809.0401](https://arxiv.org/abs/0809.0401).
-J. Borcea and P. Brändén, [“Multivariate Pólya–Schur classification problems
-in the Weyl algebra,”](https://doi.org/10.1112/plms/pdp049) *Proceedings of
-the London Mathematical Society* 101 (2010), 73–104,
-[arXiv:math/0606360](https://arxiv.org/abs/math/0606360). See also P. Brändén,
-“Unimodality, log-concavity, real-rootedness and beyond,”
-[arXiv:1410.6601](https://arxiv.org/abs/1410.6601).
+See [@BorceaBranden2009I; @BorceaBranden2010; @Branden2015].
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/RealStability.lean b/RealRooted/Challenges/RealStability.lean
index 64a173db1..612ad1c7c 100644
--- a/RealRooted/Challenges/RealStability.lean
+++ b/RealRooted/Challenges/RealStability.lean
@@ -22,6 +22,15 @@ version = 1
 section = "theorems"
 slug = "real-stability"
 formalized = "2026-10-12"
+references = [
+  "ChoeOxleySokalWagner2004",
+  "Branden2007",
+  "BorceaBranden2008MixedDeterminants",
+  "BorceaBrandenLiggett2008",
+  "MarcusSpielmanSrivastava2015II",
+  "BrandenHuh2020",
+  "Garding1959",
+]
 authors = ["Gårding", "Choe", "Oxley", "Sokal", "Wagner", "Brändén", "Borcea",
   "Liggett", "Marcus", "Spielman", "Srivastava", "Huh"]
 years = [1959, 2004, 2007, 2008, 2009, 2015, 2020]
@@ -176,7 +185,8 @@ $P \neq 0$. A polynomial with real coefficients is **real stable** if it is
 stable as a complex polynomial. It is **multiaffine** if it has degree at most
 one in each variable.
 
-**Theorem** (Brändén). A multiaffine $P \in \mathbb{R}[z_1, \dotsc, z_n]$ is
+**Theorem** (Brändén). [@Branden2007] A multiaffine
+$P \in \mathbb{R}[z_1, \dotsc, z_n]$ is
 real stable if and only if $P \neq 0$ and
 $$
 \frac{\partial P}{\partial z_i}(x) \frac{\partial P}{\partial z_j}(x) -
@@ -185,7 +195,8 @@ $$
 for all $i$, $j$ and all $x \in \mathbb{R}^n$ (the Rayleigh property).
 No homogeneity is needed.
 
-**Corollary** (Borcea–Brändén–Liggett). Let $\mu$ be a weight function on the
+**Corollary** (Borcea–Brändén–Liggett). [@BorceaBrandenLiggett2008] Let $\mu$
+be a weight function on the
 subsets of $\{1, \dotsc, n\}$ whose generating polynomial
 $\sum_S \mu(S) \prod_{i \in S} z_i$ is real stable. Then for $i \neq j$,
 $$
@@ -194,13 +205,15 @@ $$
 For a probability measure this is pairwise negative correlation; it is the
 Rayleigh inequality at the all-ones point.
 
-**Theorem** (Borcea–Brändén). Let $A$ be a Hermitian $m \times m$ matrix and
+**Theorem** (Borcea–Brändén). [@BorceaBranden2008MixedDeterminants] Let $A$
+be a Hermitian $m \times m$ matrix and
 let $B_1, \dotsc, B_n$ be positive semidefinite $m \times m$ matrices. Then
 $\det(A + z_1 B_1 + \dotsb + z_n B_n)$ is stable or identically zero. If all
 matrices are real and the determinant is not identically zero, then it is
 real stable.
 
-**Corollary** (Choe–Oxley–Sokal–Wagner). If a finite graph $G$ has a
+**Corollary** (Choe–Oxley–Sokal–Wagner). [@ChoeOxleySokalWagner2004] If a
+finite graph $G$ has a
 spanning tree, its spanning-tree polynomial
 $\sum_T \prod_{e \in T} z_e$ is real stable. By Kirchhoff's weighted
 matrix-tree theorem it equals the determinant of the reduced Laplacian
@@ -213,7 +226,7 @@ real-rooted for every $x \in \mathbb{R}^n$.
 **Theorem.** A homogeneous $P \in \mathbb{R}[z_1, \dotsc, z_n]$ is real stable
 if and only if it is hyperbolic in every direction $e$ with all $e_i > 0$.
 
-**Theorem** (Gårding). If a homogeneous $P$ is hyperbolic in the direction
+**Theorem** (Gårding). [@Garding1959] If a homogeneous $P$ is hyperbolic in the direction
 $e$, its hyperbolicity cone, the connected component of $e$ in
 $\{x : P(x) \neq 0\}$, is convex.
 
@@ -233,7 +246,8 @@ then so is this inversion. Without further variables: if $p$ has no zeros in
 the open upper half-plane and $\deg p \leq d$, then neither has
 $z^d p(-1/z)$.
 
-**Theorem** (Marcus–Spielman–Srivastava). Let $A_1, \dotsc, A_m$ be real positive
+**Theorem** (Marcus–Spielman–Srivastava).
+[@MarcusSpielmanSrivastava2015II] Let $A_1, \dotsc, A_m$ be real positive
 semidefinite $n \times n$ matrices with $\sum_i A_i = I$ and $\operatorname{tr} A_i \leq
 \varepsilon$, where $\varepsilon > 0$. Then every root of the mixed characteristic polynomial
 $$
@@ -243,7 +257,8 @@ is at most $(1 + \sqrt{\varepsilon})^2$. This is the root bound behind the solut
 Kadison–Singer problem. The formal proof follows the barrier argument, with the
 Helton–Vinnikov step replaced by an elementary Pick-function lemma.
 
-Theorem 4.1 of Marcus–Spielman–Srivastava identifies the expected characteristic
+Theorem 4.1 of Marcus–Spielman–Srivastava
+[@MarcusSpielmanSrivastava2015II, Thm. 4.1] identifies the expected characteristic
 polynomial of a sum of independent random rank-one matrices with the mixed
 characteristic polynomial of their second-moment matrices. This is the bridge
 from random vector sums to the deterministic polynomial controlled by the MSS
@@ -278,7 +293,8 @@ itself: $\operatorname{Im}(-1/z) = \operatorname{Im} z / |z|^2$. For
 $z_0 \neq 0$ the inverted polynomial evaluates to $z_0^d$ times a value of
 $P$ at a point in the product of upper half-planes.
 
-**Theorem (Brändén).** The support of a real stable polynomial is a jump system: for
+**Theorem (Brändén).** [@Branden2007] The support of a real stable polynomial
+is a jump system: for
 support points $\alpha, \beta$ and a unit step $s = \pm e_i$ from $\alpha$ towards
 $\beta$, either $\alpha + s$ is in the support, or some further unit step $t$ towards
 $\beta$ has $\alpha + s + t$ in the support. Partial derivatives, inversion in one
@@ -286,35 +302,16 @@ variable, truncation and lowest weighted-homogeneous parts preserve stability; t
 the claim to a box between $\alpha$ and $\beta$, where a homogeneous part of degree at least
 three would vanish somewhere in the product of upper half-planes.
 
-**Theorem** (Brändén–Huh). A homogeneous real stable polynomial with
+**Theorem** (Brändén–Huh). [@BrandenHuh2020] A homogeneous real stable polynomial with
 nonnegative coefficients is Lorentzian.
 
 ## References
 
-L. Gårding, [“An inequality for hyperbolic polynomials,”](https://doi.org/10.1512/iumj.1959.8.58061)
-*J. Math. Mech.* 8 (1959), 957–965. Y.-B. Choe, J. G. Oxley, A. D. Sokal and
-D. G. Wagner, [“Homogeneous multivariate polynomials with the half-plane
-property,”](https://doi.org/10.1016/S0196-8858(03)00078-2) *Adv. Appl. Math.*
-32 (2004), 88–187. P. Brändén, [“Polynomials with the half-plane property and
-matroid theory,”](https://doi.org/10.1016/j.aim.2007.05.011) *Adv. Math.* 216
-(2007), 302–320. J. Borcea and P. Brändén, [“Applications of stable
-polynomials to mixed determinants: Johnson's conjectures, unimodality, and
-symmetrized Fischer products,”](https://doi.org/10.1215/00127094-2008-018)
-*Duke Math. J.* 143 (2008), 205–223. J. Borcea, P. Brändén and T. Liggett,
-[“Negative dependence and the geometry of polynomials,”](https://doi.org/10.1090/S0894-0347-08-00618-8)
-*J. Amer. Math. Soc.* 22 (2009), 521–567. A. Marcus, D. A. Spielman and
-N. Srivastava, [“Interlacing families II: Mixed characteristic polynomials
-and the Kadison–Singer problem,”](https://doi.org/10.4007/annals.2015.182.1.8)
-*Ann. of Math.* 182 (2015), 327–350,
-[arXiv:1306.3969](https://arxiv.org/abs/1306.3969). P. Brändén and J. Huh,
-[“Lorentzian polynomials,”](https://doi.org/10.4007/annals.2020.192.3.4)
-*Ann. of Math.* 192 (2020), 821–891. See also the
-[stable polynomials](
-https://www.symmetricfunctions.com/stablePolynomials.htm#stablePolynomialDefinition)
-and the
-[Rayleigh property](
-https://www.symmetricfunctions.com/stablePolynomials.htm#stronglyRayleighProperty)
-on symmetricfunctions.com.
+See [@Garding1959; @ChoeOxleySokalWagner2004; @Branden2007;
+@BorceaBranden2008MixedDeterminants; @BorceaBrandenLiggett2008;
+@MarcusSpielmanSrivastava2015II; @BrandenHuh2020]. See also the
+[stable polynomials](sf:stablePolynomialDefinition) and the
+[Rayleigh property](sf:stronglyRayleighProperty) on symmetricfunctions.com.
 <!-- /realrooted-catalog-content -->
 
 This module exposes the multiaffine Rayleigh criterion, determinantal
diff --git a/RealRooted/Challenges/ReciprocalShift.lean b/RealRooted/Challenges/ReciprocalShift.lean
index 9af457dc7..53bb1a73d 100644
--- a/RealRooted/Challenges/ReciprocalShift.lean
+++ b/RealRooted/Challenges/ReciprocalShift.lean
@@ -12,6 +12,9 @@ version = 1
 section = "theorems"
 slug = "reciprocal-shift"
 formalized = "2026-09-26"
+references = [
+  "Fisk2006x",
+]
 
 [[theorems]]
 name = "RealRooted.reciprocalShift_reverses_strictInterl"
@@ -89,9 +92,7 @@ $\operatorname{StrictInterl}(\operatorname{reciprocalShift}(D,q),
 ## References
 
 These results were promoted from proofs in the companion OEIS formalization
-project.  For background on interlacing and polynomial reversal, see S. Fisk,
-[*Polynomials, roots, and interlacing*](https://arxiv.org/abs/math/0612833),
-arXiv:math/0612833.
+project. For background on interlacing and polynomial reversal, see [@Fisk2006x].
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/RecurrenceRootWindows.lean b/RealRooted/Challenges/RecurrenceRootWindows.lean
index 864b2f790..55f73a2f3 100644
--- a/RealRooted/Challenges/RecurrenceRootWindows.lean
+++ b/RealRooted/Challenges/RecurrenceRootWindows.lean
@@ -13,6 +13,10 @@ version = 1
 section = "theorems"
 slug = "recurrence-root-windows"
 formalized = "2026-10-01"
+references = [
+  "LiuWang2007",
+  "Branden2006",
+]
 authors = ["Liu", "Wang", "Brändén"]
 years = [2006, 2007]
 
@@ -183,13 +187,7 @@ tactics `rr_row_interlaces` and `rr_row_splits` use these routes.
 
 ## References
 
-L. L. Liu and Y. Wang, [“A unified approach to polynomial sequences with only
-real zeros,”](https://doi.org/10.1016/j.aam.2006.02.003) *Advances in Applied
-Mathematics* 38 (2007), 542–560.
-
-P. Brändén, [“On linear transformations preserving the Pólya frequency
-property,”](https://doi.org/10.1090/S0002-9947-06-03856-6) *Transactions of
-the American Mathematical Society* 358 (2006), 3697–3716.
+See [@LiuWang2007; @Branden2006].
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/RootAmplitude.lean b/RealRooted/Challenges/RootAmplitude.lean
index 8900fa75e..80f18726a 100644
--- a/RealRooted/Challenges/RootAmplitude.lean
+++ b/RealRooted/Challenges/RootAmplitude.lean
@@ -15,6 +15,9 @@ version = 1
 section = "theorems"
 slug = "root-amplitude"
 formalized = "2026-09-08"
+references = [
+  "Kozlov2022",
+]
 
 [[theorems]]
 name = "RealRooted.RootAmplitude.exp_neg_pi_sq_div_three_le_abs_eval_deriv_of_root_separation"
@@ -128,14 +131,12 @@ smallest magnitude, a reciprocal-square proportion $c > 1/2$ gives the factor
 $(2c-1)/c$; the Newton-type inequality on the first three coefficients
 specializes this to the one-fifth bound. The abstract square-sum result gives
 the related estimate for any finite family of negative factors.
+
 ## References
 
 These results were extracted from a formal proof that the rows of
 [OEIS A366159](https://oeis.org/A366159) are real-rooted and interlace. The
-motivating combinatorial objects are the Stirling complexes of D. N. Kozlov,
-[“Stirling complexes,”](https://doi.org/10.1007/s41468-022-00096-4)
-*Journal of Applied and Computational Topology* 7 (2023), 57–74,
-[arXiv:2309.17142](https://arxiv.org/abs/2309.17142).
+motivating combinatorial objects are the Stirling complexes [@Kozlov2022].
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/SchurCohn.lean b/RealRooted/Challenges/SchurCohn.lean
index a863ad416..fd2fab4b8 100644
--- a/RealRooted/Challenges/SchurCohn.lean
+++ b/RealRooted/Challenges/SchurCohn.lean
@@ -8,6 +8,10 @@ version = 1
 section = "theorems"
 slug = "schur-cohn"
 formalized = "2026-10-07"
+references = [
+  "Schur1917",
+  "Cohn1922",
+]
 authors = ["Schur", "Cohn"]
 years = [1917, 1922]
 
@@ -39,7 +43,7 @@ T p(z) = \frac{\overline{a_n}\, p(z) - a_0\, p^*(z)}{z},
 $$
 a polynomial of degree at most $n - 1$.
 
-**Theorem** (Schur–Cohn). All zeros of $p$ lie in the open unit disk if and
+**Theorem** (Schur–Cohn). [@Schur1917; @Cohn1922] All zeros of $p$ lie in the open unit disk if and
 only if $|a_0| < |a_n|$ and all zeros of $T p$ lie in the open unit disk.
 
 Iterating the transform decides disk stability in $n$ steps; this is the
@@ -55,11 +59,7 @@ has no zero with $|z| \geq 1$ when $T p$ is stable.
 
 ## References
 
-I. Schur, [“Über Potenzreihen, die im Innern des Einheitskreises beschränkt
-sind,”](https://doi.org/10.1515/crll.1917.147.205) *J. Reine Angew. Math.* 147
-(1917), 205–232; A. Cohn, [“Über die Anzahl der Wurzeln einer algebraischen
-Gleichung in einem Kreise,”](https://doi.org/10.1007/BF01215894) *Math. Z.* 14
-(1922), 110–148.
+See [@Schur1917; @Cohn1922].
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in
diff --git a/RealRooted/Challenges/SeparablePermutations.lean b/RealRooted/Challenges/SeparablePermutations.lean
index f597d799d..8e1b1d66d 100644
--- a/RealRooted/Challenges/SeparablePermutations.lean
+++ b/RealRooted/Challenges/SeparablePermutations.lean
@@ -9,6 +9,10 @@ version = 1
 section = "theorems"
 slug = "separable-permutations"
 formalized = "2026-10-09"
+references = [
+  "Zhang2026SeparableInterlacing",
+  "FuLinZeng2018",
+]
 authors = ["Zhanhe Zhang", "Fu", "Lin", "Zeng"]
 years = [2018, 2026]
 
@@ -80,9 +84,11 @@ label = "Fu–Lin–Zeng's cubic for the gamma-polynomials (via Lagrange inversi
 
 A permutation is *separable* if it avoids the patterns $2413$ and $3142$.  Let
 $S_n(t) = \sum_\pi t^{\operatorname{des}\pi}$ be the descent enumerator of the separable
-permutations of $[n]$.  Its gamma-vector is given by a convolution recurrence of Fu, Lin
-and Zeng, and $S_n(t) = (1+t)^{n-1}\,\Gamma_n\bigl(t/(1+t)^2\bigr)$ for the gamma-polynomial
-$\Gamma_n$.  Zhang represents $\Gamma_{N+2}$ as a factorial compression of the polynomial
+permutations of $[n]$. Its gamma-vector is given by a convolution recurrence of Fu, Lin
+and Zeng [@FuLinZeng2018], and
+$S_n(t) = (1+t)^{n-1}\,\Gamma_n\bigl(t/(1+t)^2\bigr)$ for the gamma-polynomial
+$\Gamma_n$. Zhang [@Zhang2026SeparableInterlacing] represents $\Gamma_{N+2}$ as a
+factorial compression of the polynomial
 $B_N$, where $B_0 = 1$ and $B_{N+1} = (x + \tfrac{N+2}{2})B_N + x B_N'$.
 
 **Theorem.** For $n \ge 2$, the polynomials $\Gamma_n$ and $\Gamma_{n+1}$ strictly interlace
@@ -111,14 +117,9 @@ the central root $-1$ is excluded by the evaluation formula for the gamma transf
 
 ## References
 
-Z. Zhang, [“Strict interlacing of the descent polynomials of separable
-permutations,”](https://doi.org/10.2139/ssrn.7510941) SSRN preprint (2026).
-S. Fu, Z. Lin and J. Zeng, [“On two unimodal descent
-polynomials,”](https://doi.org/10.1016/j.disc.2018.06.010) *Discrete
-Mathematics* 341 (2018), 2616–2626,
-[arXiv:1507.05184](https://arxiv.org/abs/1507.05184). The Lean development follows the draft
-formalization of PR #1132 by yyou59548-design. See also the
-[separable-permutation descent theorem on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedWords.htm#separableDescentRealRooted).
+See [@Zhang2026SeparableInterlacing; @FuLinZeng2018]. The Lean development
+follows the draft formalization of PR #1132 by yyou59548-design. See also the
+[separable-permutation descent theorem on symmetricfunctions.com](sf:separableDescentRealRooted).
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in `RealRooted.SeparablePermutations` and
diff --git a/RealRooted/Challenges/SmallInterlacingMatrices.lean b/RealRooted/Challenges/SmallInterlacingMatrices.lean
index 2133e6fc1..7ba75a2f3 100644
--- a/RealRooted/Challenges/SmallInterlacingMatrices.lean
+++ b/RealRooted/Challenges/SmallInterlacingMatrices.lean
@@ -8,6 +8,10 @@ version = 1
 section = "theorems"
 slug = "small-interlacing-matrices"
 formalized = "2026-10-06"
+references = [
+  "Fisk2006x",
+  "Branden2015",
+]
 authors = ["Fisk", "Brändén"]
 years = [2006, 2015]
 
@@ -87,7 +91,7 @@ In particular, exactly 18 nondegenerate matrices preserve interlacing.
 
 ## Proof idea
 
-By the matrix criterion of Brändén (Theorem 7.8.5), a nonnegative polynomial
+By the matrix criterion of Brändén [@Branden2015, Thm. 7.8.5], a nonnegative polynomial
 matrix preserves interlacing sequences exactly when each ordered
 $2 \times 2$ submatrix, repeated indices allowed, satisfies an affine
 interlacing condition. For entries in $\{0, 1, x\}$ these nine conditions
@@ -98,11 +102,8 @@ follow by deciding the finite table.
 
 ## References
 
-S. Fisk, *Polynomials, roots, and interlacing*, arXiv:math/0612833, Chapter 3;
-P. Brändén, “Unimodality, log-concavity, real-rootedness and beyond,”
-*Handbook of Enumerative Combinatorics*, CRC Press, 2015, Theorem 7.8.5.
-See the
-[table on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedInterlacing.htm#smallInterlacingMatrices).
+See [@Fisk2006x, Ch. 3] and [@Branden2015, Thm. 7.8.5]. See the
+[table on symmetricfunctions.com](sf:smallInterlacingMatrices).
 <!-- /realrooted-catalog-content -->
 
 This module exposes the checked classification of the atomic `2 × 2`
diff --git a/RealRooted/Challenges/Sturm.lean b/RealRooted/Challenges/Sturm.lean
index 788c11472..3842f5317 100644
--- a/RealRooted/Challenges/Sturm.lean
+++ b/RealRooted/Challenges/Sturm.lean
@@ -10,6 +10,9 @@ version = 1
 section = "theorems"
 slug = "sturm-root-counting"
 formalized = "2026-09-27"
+references = [
+  "Sylvester1853",
+]
 authors = ["Sturm", "Hermite", "Sylvester"]
 years = [1829, 1853]
 
@@ -77,7 +80,7 @@ distinct real roots in the open interval.
 Repeated factors are removed by a canonical squarefree normalization. If the
 interval contains every root, the same count tests whether all roots are real.
 
-**Hermite–Sylvester criterion.** Let $f$ be a monic real polynomial of degree
+**Hermite–Sylvester criterion.** [@Sylvester1853] Let $f$ be a monic real polynomial of degree
 $n$, and let $s_k = \sum_z z^k$ be the Newton power sums of its complex roots,
 counted with multiplicity. The Hermite matrix is
 $H_f = (s_{i+j})_{0 \le i, j \le n-1}$. Then $f$ is real-rooted if and only if
@@ -89,19 +92,15 @@ if its cubic discriminant is nonnegative.
 
 ## References
 
-J. C. F. Sturm, “Mémoire sur la résolution des équations numériques,”
-*Bulletin des Sciences de Férussac* 11 (1829), 419–422;
-C. Hermite, “Remarques sur le théorème de M. Sturm,” *Comptes rendus de
-l’Académie des sciences* 36 (1853), 52–54;
-J. J. Sylvester, [“On a theory of the syzygetic relations of two rational
-integral functions,”](https://doi.org/10.1098/rstl.1853.0018) *Philosophical
-Transactions of the Royal Society of London* 143 (1853), 407–548. See also
-the root-counting
-overview on [symmetricfunctions.com][sturm-overview] and the
+J. C. F. Sturm, “Mémoire sur la résolution des équations numériques,” *Bulletin
+des Sciences de Férussac* 11 (1829), 419–422; C. Hermite, “Remarques sur le
+théorème de M. Sturm,” *Comptes rendus de l’Académie des sciences* 36 (1853),
+52–54. See [@Sylvester1853], the root-counting overview on
+[symmetricfunctions.com][sturm-overview], and the
 [Hermite–Sylvester theorem][hermite-sylvester] there.
 
-[sturm-overview]: https://www.symmetricfunctions.com/realRootedInterlacing.htm#sturmRootCounting
-[hermite-sylvester]: https://www.symmetricfunctions.com/realRooted.htm#hermiteSylvesterTheorem
+[sturm-overview]: sf:sturmRootCounting
+[hermite-sylvester]: sf:hermiteSylvesterTheorem
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in
diff --git a/RealRooted/Challenges/SturmSequenceFamilies.lean b/RealRooted/Challenges/SturmSequenceFamilies.lean
index cdb6348f4..35a8be47d 100644
--- a/RealRooted/Challenges/SturmSequenceFamilies.lean
+++ b/RealRooted/Challenges/SturmSequenceFamilies.lean
@@ -15,6 +15,14 @@ version = 1
 section = "families"
 slug = "sturm-sequence-families"
 formalized = "2026-05-12"
+references = [
+  "GesselStanley1978",
+  "MantaciRakotondrajao2003",
+  "Bona2009",
+  "ChowShiu2011",
+  "Wang2014",
+  "BonaHezo2016",
+]
 authors = ["Gessel", "Stanley", "Bóna", "Mantaci", "Rakotondrajao", "Chow", "Shiu", "Mező",
   "Wang"]
 years = [1978, 2003, 2009, 2011, 2014, 2016]
@@ -172,31 +180,12 @@ nonnegative combinations of common interlacers handles the extra term.
 
 ## References
 
-I. Gessel and R. P. Stanley, [“Stirling
-polynomials,”](https://doi.org/10.1016/0097-3165(78)90042-0) *Journal of
-Combinatorial Theory, Series A* 24 (1978), 24–33.
-R. Mantaci and F. Rakotondrajao, [“Exceedingly
-deranging!”](https://doi.org/10.1016/s0196-8858(02)00531-6) *Advances in
-Applied Mathematics* 30 (2003), 177–188.
-M. Bóna, [“Real zeros and normal distribution for statistics on Stirling
-permutations defined by Gessel and Stanley,”](https://doi.org/10.1137/070702254)
-*SIAM Journal on Discrete Mathematics* 23 (2009), 401–406.
-C.-O. Chow and W. C. Shiu, [“Counting simsun permutations by
-descents,”](https://doi.org/10.1007/s00026-011-0113-6) *Annals of
-Combinatorics* 15 (2011), 625–635.
-D. G. L. Wang, [“On colored set partitions of type
-$B_n$,”](https://doi.org/10.2478/s11533-014-0419-9) *Central European Journal
-of Mathematics* 12 (2014), 1372–1381.
-M. Bóna and I. Mező, [“Real zeros and partitions without singleton
-blocks,”](https://doi.org/10.1016/j.ejc.2015.07.021) *European Journal of
-Combinatorics* 51 (2016), 500–510.
-See also the
-[Touchard polynomials](https://www.symmetricfunctions.com/realRootedWords.htm#touchardPolynomial),
-[Stirling permutations](https://www.symmetricfunctions.com/realRootedWords.htm#stirlingPermutationDescentPolynomial),
-the
-[derangement polynomials](https://www.symmetricfunctions.com/realRootedWords.htm#derangementPolynomial),
-and the
-[Motzkin-polynomial theorem](https://www.symmetricfunctions.com/realRootedCatalan.htm#motzkinPolynomialsRealRooted)
+See [@GesselStanley1978; @MantaciRakotondrajao2003; @Bona2009;
+@ChowShiu2011; @Wang2014; @BonaHezo2016]. See also the
+[Touchard polynomials](sf:touchardPolynomial),
+[Stirling permutations](sf:stirlingPermutationDescentPolynomial), the
+[derangement polynomials](sf:derangementPolynomial), and the
+[Motzkin-polynomial theorem](sf:motzkinPolynomialsRealRooted)
 on symmetricfunctions.com.
 <!-- /realrooted-catalog-content -->
 
diff --git a/RealRooted/Challenges/SzaszGrowth.lean b/RealRooted/Challenges/SzaszGrowth.lean
index bbfaf0c14..479a1e013 100644
--- a/RealRooted/Challenges/SzaszGrowth.lean
+++ b/RealRooted/Challenges/SzaszGrowth.lean
@@ -10,6 +10,9 @@ version = 1
 section = "theorems"
 slug = "szasz-growth"
 formalized = "2026-10-13"
+references = [
+  "BorceaBranden2009I",
+]
 authors = ["Borcea", "Brändén"]
 years = [2009]
 
@@ -87,7 +90,8 @@ $$
 This applies to a polynomial $\prod_j (1 + \xi_j z)$ with no zeros in the open upper half-plane.
 
 In the next two theorems, assume that $a(0) = 1$ and that $J_\beta f$ is stable for every
-$\beta \in \mathbb N^\sigma$. (For stable $f$ this is Borcea–Brändén, Lemma 5.2.)
+$\beta \in \mathbb N^\sigma$. For stable $f$ this is Borcea–Brändén
+[@BorceaBranden2009I, Lemma 5.2].
 
 **Theorem** (coefficient bound). For every $\beta$,
 $$
@@ -144,13 +148,7 @@ general support.
 
 ## References
 
-J. Borcea and P. Brändén,
-[“The Lee–Yang and Pólya–Schur programs. I. Linear operators preserving
-stability,”](https://arxiv.org/abs/0809.0401) *Invent. Math.* 177
-(2009), 541–569,
-[doi:10.1007/s00222-009-0189-3](https://doi.org/10.1007/s00222-009-0189-3),
-[arXiv:0809.0401](https://arxiv.org/abs/0809.0401), Lemmas 5.2 and 5.4 and
-Theorems 5.5 and 5.6.
+See [@BorceaBranden2009I, Lemmas 5.2 and 5.4 and Thms. 5.5 and 5.6].
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade. The univariate estimate is in
diff --git a/RealRooted/Challenges/ThresholdMatrices.lean b/RealRooted/Challenges/ThresholdMatrices.lean
index 738635cfd..26d9fbd7c 100644
--- a/RealRooted/Challenges/ThresholdMatrices.lean
+++ b/RealRooted/Challenges/ThresholdMatrices.lean
@@ -9,6 +9,11 @@ version = 1
 section = "theorems"
 slug = "threshold-matrices"
 formalized = "2026-10-06"
+references = [
+  "HaglundZhang2019",
+  "GustafssonSolus2020",
+  "MaMaYeh2017x",
+]
 authors = ["Haglund", "Zhang", "Gustafsson", "Solus"]
 years = [2019, 2020]
 
@@ -54,10 +59,9 @@ A matrix of such rows with weakly increasing thresholds maps interlacing
 sequences of polynomials with nonnegative coefficients to interlacing
 sequences, as long as finitely many $2 \times 2$ compatibility checks hold.
 
-**Theorem** (Haglund–Zhang). Threshold matrices whose markers are $1$ or
+**Theorem** (Haglund–Zhang). [@HaglundZhang2019] Threshold matrices whose markers are $1$ or
 $1 + x$ preserve interlacing sequences. As a consequence, the binomial
-Eulerian polynomials are real-rooted, confirming a conjecture of Ma, Ma and
-Yeh.
+Eulerian polynomials are real-rooted, confirming a conjecture of Ma, Ma and Yeh [@MaMaYeh2017x].
 
 Here the binomial Eulerian polynomials are defined by the Haglund–Zhang
 recursion on refined vectors, with marker $1 + x$ at threshold zero and $1$ at
@@ -66,7 +70,8 @@ $\sum_k T(n,k)\,x^k$ of [OEIS A046802](https://oeis.org/A046802), the
 $h$-polynomials of the stellohedra; the first four rows are checked in Lean,
 but the identification with the closed formula for all $n$ is not formalized.
 
-**Theorem** (Gustafsson–Solus, Lemma 3.4). Row-threshold matrices, in which
+**Theorem** (Gustafsson–Solus, Lemma 3.4). [@GustafssonSolus2020, Lemma 3.4]
+Row-threshold matrices, in which
 some rows drop their pivot entry, preserve interlacing sequences when the
 thresholds increase weakly and no switch follows a dropped pivot.
 Gustafsson and Solus use this to prove that the box polynomials of
@@ -75,16 +80,8 @@ real-rooted.
 
 ## References
 
-J. Haglund and P. B. Zhang, [“Real-rootedness of variations of Eulerian
-polynomials,”](https://doi.org/10.1016/j.aam.2019.05.001) *Advances in Applied
-Mathematics* 109 (2019), 38–54,
-[arXiv:1902.01278](https://arxiv.org/abs/1902.01278). N. Gustafsson and
-L. Solus, [“Derangements, Ehrhart theory, and local
-$h$-polynomials,”](https://doi.org/10.1016/j.aim.2020.107169) *Advances in
-Mathematics* 369 (2020), 107169. J. Ma, S.-M. Ma and Y.-N. Yeh, “Recurrence
-relations for binomial-Eulerian polynomials,”
-[arXiv:1711.09016](https://arxiv.org/abs/1711.09016) (2017). See also the
-[binomial Eulerian polynomial on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedWords.htm#binomialEulerianPolynomial).
+See [@HaglundZhang2019; @GustafssonSolus2020; @MaMaYeh2017x]. See also the
+[binomial Eulerian polynomial on symmetricfunctions.com](sf:binomialEulerianPolynomial).
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in
diff --git a/RealRooted/Challenges/ToricContribution.lean b/RealRooted/Challenges/ToricContribution.lean
index c17823744..ccc16e5bc 100644
--- a/RealRooted/Challenges/ToricContribution.lean
+++ b/RealRooted/Challenges/ToricContribution.lean
@@ -9,6 +9,9 @@ version = 1
 section = "theorems"
 slug = "toric-g-contributions"
 formalized = "2026-09-02"
+references = [
+  "Xiao2026x",
+]
 authors = ["Xiao"]
 years = [2026]
 
@@ -76,7 +79,8 @@ Xiao conjectured in Conjecture 4.2 that, for each
 $n = 2m + \varepsilon$ with $\varepsilon \in \{0, 1\}$,
 the row $(g_{n,0}, g_{n,1}, \dotsc, g_{n,\lfloor n/2\rfloor})$ is an interlacing sequence.
 
-**Theorem (Xiao's conjecture).** For all $m$ and $\varepsilon \leq 1$, the toric
+**Theorem (Xiao's conjecture).** [@Xiao2026x] For all $m$ and
+$\varepsilon \leq 1$, the toric
 contribution row is an interlacing sequence.
 
 The library indexes by the offset from the middle of the row:
@@ -89,11 +93,10 @@ reversed contributions is real-rooted.
 
 ## References
 
-Q. Xiao, [“The real-rootedness of the toric g-contribution
-polynomials,”](https://arxiv.org/abs/2609.01086) arXiv:2609.01086 (2026).
-Common interleavers are described on the
+See [@Xiao2026x]. Common interleavers are described on the
 [common interleaver page](/RealRooted/concepts/common-interleavers/). See also
-the [toric $g$-contribution polynomial on symmetricfunctions.com](https://www.symmetricfunctions.com/polytopes.htm#toricGContributionPolynomial).
+the [toric $g$-contribution polynomial](sf:toricGContributionPolynomial)
+on symmetricfunctions.com.
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.  The proofs live in
diff --git a/RealRooted/Challenges/TotallyNonnegativeHadamardObstruction.lean b/RealRooted/Challenges/TotallyNonnegativeHadamardObstruction.lean
index 0b7e65434..c9247ffd9 100644
--- a/RealRooted/Challenges/TotallyNonnegativeHadamardObstruction.lean
+++ b/RealRooted/Challenges/TotallyNonnegativeHadamardObstruction.lean
@@ -8,6 +8,9 @@ version = 1
 section = "theorems"
 slug = "totally-nonnegative-hadamard-obstruction"
 formalized = "2026-09-30"
+references = [
+  "Wagner1992",
+]
 
 [[theorems]]
 name = "RealRooted.TotallyNonnegativeHadamardObstruction.exists_totallyNonneg_hadamard_det_neg"
@@ -34,10 +37,8 @@ fails; see `RealRooted.not_hurwitz_schurProduct_det_fin_three_nonneg`.
 
 ## References
 
-D. G. Wagner, [“Total positivity of Hadamard
-products,”](https://doi.org/10.1016/0022-247X(92)90261-B) *Journal of
-Mathematical Analysis and Applications* 163 (1992), 459–483. See also the
-[Hadamard product on symmetricfunctions.com](https://www.symmetricfunctions.com/realRooted.htm#hadamardProductDefinition).
+See [@Wagner1992] and the
+[Hadamard product on symmetricfunctions.com](sf:hadamardProductDefinition).
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/TotallyNonnegativeMatrices.lean b/RealRooted/Challenges/TotallyNonnegativeMatrices.lean
index 2acf1c8da..0388ec144 100644
--- a/RealRooted/Challenges/TotallyNonnegativeMatrices.lean
+++ b/RealRooted/Challenges/TotallyNonnegativeMatrices.lean
@@ -14,6 +14,12 @@ version = 1
 section = "theorems"
 slug = "totally-nonnegative-matrices"
 formalized = "2026-09-05"
+references = [
+  "Fisk2006x",
+  "Cryer1976",
+  "Karlin1968",
+  "FallatJohnson2011",
+]
 authors = ["Gantmacher", "Krein", "Karlin", "Cryer", "Fisk"]
 years = [1937, 1968, 1976, 2006]
 
@@ -114,7 +120,8 @@ are positive (the oscillatory case), then both characteristic polynomials of
 $A$ and $A'$ have simple roots, they have no common root, the interlacing is
 strict, and all eigenvalues of $A$ are positive.
 
-**Theorem** (TNN mixing, Fisk). Let $f_1, \dotsc, f_n$ be real-rooted
+**Theorem** (TNN mixing, Fisk). [@Fisk2006x, Thm. 3.7] Let
+$f_1, \dotsc, f_n$ be real-rooted
 polynomials with positive leading coefficients that form an interlacing
 sequence, so $f_i \ll f_j$ for $i < j$, and let $A$ be a TNN $m \times n$
 matrix. Then the nonzero entries of $A (f_1, \dotsc, f_n)^{\mathsf T}$ are
@@ -125,7 +132,7 @@ sequence in row order.
 $c \in \mathbb{R}^n$, then $\operatorname{var}(Ac) \leq \operatorname{var}(c)$,
 where $\operatorname{var}$ counts sign changes after deleting zero entries.
 
-**Theorem** (Cryer). A nonsingular lower-triangular real matrix is TNN as soon
+**Theorem** (Cryer). [@Cryer1976] A nonsingular lower-triangular real matrix is TNN as soon
 as every minor formed from arbitrary rows and an initial segment of the
 columns is nonnegative.
 
@@ -133,7 +140,7 @@ columns is nonnegative.
 matrix $A$ is primitive, then the eigenvalues of $A$ are real, positive and
 simple. Karlin's Gaussian matrices $\bigl(e^{-a(i-j)^2}\bigr)$ with $a > 0$ are
 strictly totally positive: every minor with strictly increasing rows and
-columns is positive.
+columns is positive [@Karlin1968].
 
 ## Proof idea
 
@@ -161,18 +168,11 @@ and passing to the limit.
 
 F. R. Gantmacher and M. G. Krein, *Oscillation Matrices and Kernels and Small
 Vibrations of Mechanical Systems*, 1937; revised English edition, AMS Chelsea,
-2002. S. Karlin, *Total Positivity*, Vol. I, Stanford University Press, 1968.
-C. W. Cryer, “Some properties of totally positive matrices,” *Linear Algebra
-Appl.* 15 (1976), 1–25,
-[DOI 10.1016/0024-3795(76)90076-8](https://doi.org/10.1016/0024-3795(76)90076-8).
-S. M. Fallat and C. R. Johnson, *Totally Nonnegative
-Matrices*, Princeton University Press, 2011. S. Fisk, [*Polynomials, roots,
-and interlacing*](https://arxiv.org/abs/math/0612833), arXiv:math/0612833
-(2006), Theorem 3.7. See the
-[TNN matrices](https://www.symmetricfunctions.com/polyaFrequency.htm#totallyNonnegativeMatrix),
-[variation-diminishing property](https://www.symmetricfunctions.com/polyaFrequency.htm#variationDiminishingProperty)
-and [TNN mixing](https://www.symmetricfunctions.com/realRootedInterlacing.htm#tnnMixingInterlacing)
-entries on symmetricfunctions.com.
+2002. See [@Karlin1968; @Cryer1976; @FallatJohnson2011]. TNN mixing appears
+in [@Fisk2006x, Thm. 3.7]. See the
+[TNN matrices](sf:totallyNonnegativeMatrix),
+[variation-diminishing property](sf:variationDiminishingProperty), and
+[TNN mixing](sf:tnnMixingInterlacing) entries on symmetricfunctions.com.
 <!-- /realrooted-catalog-content -->
 
 This module exposes the spectral, interlacing, mixing, sign-variation and
diff --git a/RealRooted/Challenges/UniformMatroidHodgePoincare.lean b/RealRooted/Challenges/UniformMatroidHodgePoincare.lean
index 7294bd036..0708ed6ab 100644
--- a/RealRooted/Challenges/UniformMatroidHodgePoincare.lean
+++ b/RealRooted/Challenges/UniformMatroidHodgePoincare.lean
@@ -9,6 +9,12 @@ version = 1
 section = "theorems"
 slug = "uniform-matroid-hodge-poincare"
 formalized = "2026-10-09"
+references = [
+  "BinderVecchi2026x",
+  "Athanasiadis2025EulerianTransformation",
+  "BrandenJochemko2022",
+  "BrandenSolus2019",
+]
 authors = ["Binder", "Vecchi", "Athanasiadis", "Brändén", "Jochemko", "Solus"]
 years = [2021, 2022, 2025, 2026]
 
@@ -74,7 +80,8 @@ $$
 where $d_j$ are the derangement polynomials, $A_s$ the Eulerian polynomials, and
 $W_{j,i}(r,n) = \binom nj \sum_{\ell=1}^{n-r-i+1}\binom{n-j-\ell}{r-j-1}\binom{n-r-\ell}{i-1}$.
 
-**Theorem** (Binder–Vecchi, Theorems 1.4 and 1.6). Both polynomials are real-rooted.
+**Theorem** (Binder–Vecchi, Theorems 1.4 and 1.6).
+[@BinderVecchi2026x, Thms. 1.4 and 1.6] Both polynomials are real-rooted.
 
 ## Proof idea
 
@@ -84,7 +91,8 @@ the Eulerian transformation $\mathcal A\colon 1 \mapsto 1,\ x^n \mapsto xA_n(x)$
 Brändén–Jochemko. A Vandermonde-type identity shows that $g$ has nonnegative coordinates in
 the magic basis $x^p(1+x)^{r-1-p}$. Both transforms send such polynomials to real-rooted ones:
 for $\mathcal D$ this is Brändén–Solus (via the Brändén–Vecchi Chow resolution of the Pascal
-matrix), and for $\mathcal A$ it is Athanasiadis's theorem, proved here with the Brändén–Vecchi
+matrix), and for $\mathcal A$ it is Athanasiadis's theorem
+[@Athanasiadis2025EulerianTransformation], proved here with the Brändén–Vecchi
 staircase row transform.
 
 For $i = 0$ the two families are the Chow and augmented Chow polynomials of $U_{r,n}$; their
@@ -96,16 +104,8 @@ about the explicit formulas.
 
 ## References
 
-K. Binder and L. Vecchi, “Augmented singular cohomology, uniform matroids,
-and real-rootedness,”
-[arXiv:2609.15946](https://arxiv.org/abs/2609.15946) (2026). C. A. Athanasiadis,
-[“On the real-rootedness of the Eulerian
-transformation,”](https://doi.org/10.1112/jlms.70083) *J. London Math. Soc.*
-111 (2025), e70083. P. Brändén and K. Jochemko, [“The Eulerian
-transformation,”](https://doi.org/10.1090/tran/8539) *Trans. Amer. Math. Soc.*
-375 (2022), 1917–1931. P. Brändén and L. Solus, [“Symmetric decompositions and
-real-rootedness,”](https://doi.org/10.1093/imrn/rnz059) *International
-Mathematics Research Notices* 2021, 7764–7798.
+See [@BinderVecchi2026x; @Athanasiadis2025EulerianTransformation;
+@BrandenJochemko2022; @BrandenSolus2019].
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade.
diff --git a/RealRooted/Challenges/UnivariatePoissonStar.lean b/RealRooted/Challenges/UnivariatePoissonStar.lean
index f560a281c..1fadbbf4c 100644
--- a/RealRooted/Challenges/UnivariatePoissonStar.lean
+++ b/RealRooted/Challenges/UnivariatePoissonStar.lean
@@ -12,6 +12,9 @@ version = 1
 section = "theorems"
 slug = "univariate-poisson-star-c"
 formalized = "2026-10-12"
+references = [
+  "Branden2006",
+]
 authors = ["Brändén"]
 years = [2006]
 
@@ -143,10 +146,7 @@ multiplication, proving both multiplicativity and associativity of $\star_c$.
 
 ## References
 
-P. Brändén, [“On linear transformations preserving the Pólya frequency
-property,”](https://doi.org/10.1090/S0002-9947-06-03856-6) *Transactions of
-the American Mathematical Society* 358 (2006), no. 8, 3697–3716,
-[arXiv:math/0403364](https://arxiv.org/abs/math/0403364).
+See [@Branden2006].
 <!-- /realrooted-catalog-content -->
 -/
 
diff --git a/RealRooted/Challenges/VanDerWaerden.lean b/RealRooted/Challenges/VanDerWaerden.lean
index f8e661d7b..f629a6b66 100644
--- a/RealRooted/Challenges/VanDerWaerden.lean
+++ b/RealRooted/Challenges/VanDerWaerden.lean
@@ -8,6 +8,11 @@ version = 1
 section = "theorems"
 slug = "van-der-waerden"
 formalized = "2026-10-07"
+references = [
+  "Gurvits2008",
+  "Egorychev1981",
+  "Falikman1981",
+]
 authors = ["Egorychev", "Falikman", "Gurvits"]
 years = [1981, 2008]
 
@@ -36,12 +41,14 @@ label = "Gurvits' univariate coefficient inequality"
 <!-- realrooted-catalog-content -->
 # The van der Waerden permanent bound
 
-**Theorem (Egorychev, Falikman).** If $A$ is an $n \times n$ doubly stochastic matrix, then
+**Theorem (Egorychev, Falikman).** [@Egorychev1981; @Falikman1981]
+If $A$ is an $n \times n$ doubly stochastic matrix, then
 $$
 \operatorname{per} A \ge \frac{n!}{n^n}.
 $$
 
-The formal proof follows Gurvits. The *capacity* of a polynomial $p(x_1, \dotsc, x_n)$ with
+The formal proof follows Gurvits [@Gurvits2008]. The *capacity* of a polynomial
+$p(x_1, \dotsc, x_n)$ with
 nonnegative coefficients is $\inf_{x > 0} p(x) / (x_1 \cdots x_n)$. For a real stable
 polynomial $p$ of total degree at most $k$ in $k$ variables, differentiating in one variable
 and setting it to zero gives a real stable polynomial whose capacity is at least
@@ -53,15 +60,8 @@ $x_1 \cdots x_n$, which is $\operatorname{per} A$, and the factors multiply to $
 
 ## References
 
-G. P. Egorychev, [“The solution of van der Waerden's problem for
-permanents,”](https://doi.org/10.1016/0001-8708(81)90044-X) *Advances in
-Mathematics* 42 (1981), 299–305. D. I. Falikman, [“Proof of the van der
-Waerden conjecture regarding the permanent of a doubly stochastic
-matrix,”](https://doi.org/10.1007/BF01163285) *Mathematical Notes* 29 (1981),
-475–479. L. Gurvits, [“Van der Waerden/Schrijver–Valiant like conjectures and
-stable (aka hyperbolic) homogeneous polynomials: one theorem for
-all,”](https://doi.org/10.37236/790) *Electronic Journal of Combinatorics* 15
-(2008), R66.
+The classical proofs are [@Egorychev1981; @Falikman1981]. The formal proof
+follows [@Gurvits2008].
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade. The proof lives in `RealRooted.Gurvits`.
diff --git a/RealRooted/Challenges/VeroneseSections.lean b/RealRooted/Challenges/VeroneseSections.lean
index 50a57eb9a..31dba890b 100644
--- a/RealRooted/Challenges/VeroneseSections.lean
+++ b/RealRooted/Challenges/VeroneseSections.lean
@@ -8,6 +8,9 @@ version = 1
 section = "theorems"
 slug = "veronese-sections"
 formalized = "2026-10-05"
+references = [
+  "AthanasiadisWagner2024x",
+]
 
 [[definitions]]
 name = "RealRooted.veroneseSectionPolynomial"
@@ -32,15 +35,13 @@ nonnegativity.
 
 ## References
 
-[C. A. Athanasiadis and D. G. Wagner, “Veronese sections and interlacing
-matrices of polynomials and formal power
-series,”](https://arxiv.org/abs/2404.12989) arXiv:2404.12989 (2024), proves a
-stronger fully interlacing statement. See also the
-[Veronese sections on symmetricfunctions.com](https://www.symmetricfunctions.com/polyaFrequency.htm#veroneseSectionsRealRooted).
+A stronger fully interlacing statement appears in [@AthanasiadisWagner2024x].
+See also the
+[Veronese sections on symmetricfunctions.com](sf:veroneseSectionsRealRooted).
 <!-- /realrooted-catalog-content -->
 
 Human statement:
-https://www.symmetricfunctions.com/polyaFrequency.htm#veroneseSectionsRealRooted
+sf:veroneseSectionsRealRooted
 
 Catalog reference: the standard PF/TNN proof via Aissen--Schoenberg--Whitney.
 
diff --git a/RealRooted/Challenges/Wagner.lean b/RealRooted/Challenges/Wagner.lean
index 5ea86a043..8532eee34 100644
--- a/RealRooted/Challenges/Wagner.lean
+++ b/RealRooted/Challenges/Wagner.lean
@@ -10,6 +10,9 @@ version = 1
 section = "theorems"
 slug = "wagner"
 formalized = "2026-07-10"
+references = [
+  "Wagner1992",
+]
 authors = ["Wagner"]
 years = [1992]
 
@@ -61,14 +64,12 @@ $F_i \ll F_j$ for all $a \le i \le j \le b$.
 
 ## References
 
-D. G. Wagner, [“Total positivity of Hadamard
-products,”](https://doi.org/10.1016/0022-247X(92)90261-B) *Journal of
-Mathematical Analysis and Applications* 163 (1992), 459–483. See the
-[contextual statement on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedInterlacing.htm#wagnerLemma).
+See [@Wagner1992] and the
+[contextual statement on symmetricfunctions.com](sf:wagnerLemma).
 <!-- /realrooted-catalog-content -->
 
 Human statement:
-https://www.symmetricfunctions.com/realRootedInterlacing.htm#wagnerLemma
+sf:wagnerLemma
 
 Original publication: D. G. Wagner, "Total positivity of Hadamard products",
 J. Math. Anal. Appl. 163 (1992), 459--483.
diff --git a/RealRooted/Challenges/WangYeh.lean b/RealRooted/Challenges/WangYeh.lean
index b39b39c18..2e4503863 100644
--- a/RealRooted/Challenges/WangYeh.lean
+++ b/RealRooted/Challenges/WangYeh.lean
@@ -8,6 +8,9 @@ version = 1
 section = "theorems"
 slug = "wang-yeh"
 formalized = "2026-10-06"
+references = [
+  "WangYeh2005",
+]
 authors = ["Wang", "Yeh"]
 years = [2005]
 
@@ -46,7 +49,8 @@ either zero or real-rooted with nonpositive zeros. A sequence is
 Pólya-frequency (`IsPolyaFreqSeq`) if its Toeplitz matrix is totally
 nonnegative.
 
-**Theorem (Wang–Yeh).** Let $f, g \in \mathbb{R}[x]$ have positive leading
+**Theorem (Wang–Yeh).** [@WangYeh2005, Thm. 1] Let $f, g \in \mathbb{R}[x]$
+have positive leading
 coefficients and satisfy $g \ll f$. If $a, b, c, d \in \mathbb{R}$ satisfy
 $bc \leq ad$, then
 $$
@@ -96,14 +100,9 @@ inequalities give the determinant condition at every step.
 
 ## References
 
-Y. Wang and Y.-N. Yeh, [“Polynomials with real zeros and Pólya frequency
-sequences,”](https://doi.org/10.1016/j.jcta.2004.07.008) *Journal of
-Combinatorial Theory, Series A* 109 (2005), 63–74, Theorem 1 and Corollaries 2
-and 3.
-See also the
-[Wang–Yeh criterion on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedInterlacing.htm#wangYehAffineCriterion)
-and the
-[Wang–Yeh PF preserver](https://www.symmetricfunctions.com/polyaFrequency.htm#wangYehPfPreserver).
+The affine criterion and PF preserver are [@WangYeh2005, Thm. 1 and Cors. 2–3].
+See also the [Wang–Yeh criterion](sf:wangYehAffineCriterion) and the
+[Wang–Yeh PF preserver](sf:wangYehPfPreserver) on symmetricfunctions.com.
 <!-- /realrooted-catalog-content -->
 
 The affine criterion lives in `RealRooted.WangYeh.Affine`, the PF preserver in
diff --git a/RealRooted/Challenges/Wronskian.lean b/RealRooted/Challenges/Wronskian.lean
index ee8f73e2e..27ce10e37 100644
--- a/RealRooted/Challenges/Wronskian.lean
+++ b/RealRooted/Challenges/Wronskian.lean
@@ -9,6 +9,9 @@ version = 1
 section = "theorems"
 slug = "wronskian-criterion"
 formalized = "2026-10-06"
+references = [
+  "Fisk2006x",
+]
 
 [[theorems]]
 name = "RealRooted.strictInterl_iff_wronskian_eval_nonneg"
@@ -143,10 +146,8 @@ $q^2 + (x - a)^2 (q'^2 - q q'')$.
 
 ## References
 
-S. Fisk, [*Polynomials, roots, and
-interlacing*](https://arxiv.org/abs/math/0612833), arXiv:math/0612833 (2006).
-See also the Wronskian criterion in the
-[interlacing overview on symmetricfunctions.com](https://www.symmetricfunctions.com/realRootedInterlacing.htm#wronskianCriterion).
+See [@Fisk2006x] and the Wronskian criterion in the
+[interlacing overview on symmetricfunctions.com](sf:wronskianCriterion).
 <!-- /realrooted-catalog-content -->
 
 This module is a catalog facade. The weak converse and the counterexample
diff --git a/scripts/build_challenge_pages.py b/scripts/build_challenge_pages.py
index 77a0a437e..469ab6f35 100644
--- a/scripts/build_challenge_pages.py
+++ b/scripts/build_challenge_pages.py
@@ -11,6 +11,8 @@
 sys.dont_write_bytecode = True
 
 from challenge_catalog import (
+    BASE_PATH,
+    SITE_ORIGIN,
     CatalogError,
     load_catalog,
     render_site,
@@ -42,6 +44,16 @@ def parse_args() -> argparse.Namespace:
         type=pathlib.Path,
         help="post-Lean audit report required for publishable output",
     )
+    parser.add_argument(
+        "--base-path",
+        default=BASE_PATH,
+        help=f"URL path at which the site is served (default: {BASE_PATH})",
+    )
+    parser.add_argument(
+        "--site-origin",
+        default=SITE_ORIGIN,
+        help=f"origin used for absolute manifest links (default: {SITE_ORIGIN})",
+    )
     return parser.parse_args()
 
 
@@ -57,7 +69,14 @@ def main() -> int:
         if args.check:
             # Render in memory so this check covers templates, internal links,
             # and optional catalog side data such as the Authors directory.
-            render_site(root, pages, resolved, revision)
+            render_site(
+                root,
+                pages,
+                resolved,
+                revision,
+                base_path=args.base_path,
+                site_origin=args.site_origin,
+            )
             print(f"ok: validated {len(pages)} curated catalog pages")
             return 0
         output = (
@@ -67,7 +86,14 @@ def main() -> int:
         )
         if output == root or output.is_relative_to(root):
             raise CatalogError("generated pages must use an output directory outside the repository")
-        files = render_site(root, pages, resolved, revision)
+        files = render_site(
+            root,
+            pages,
+            resolved,
+            revision,
+            base_path=args.base_path,
+            site_origin=args.site_origin,
+        )
         write_site(output, files)
         print(f"ok: wrote {len(files)} catalog files to {output}")
         if args.preview:
diff --git a/scripts/challenge_catalog.py b/scripts/challenge_catalog.py
index 34d27b881..6c6ec881c 100644
--- a/scripts/challenge_catalog.py
+++ b/scripts/challenge_catalog.py
@@ -28,6 +28,8 @@
 
 
 BASE_PATH = "/RealRooted/"
+SITE_ORIGIN = "https://peralexandersson.github.io"
+SYMCAT_ORIGIN = "https://www.symmetricfunctions.com/"
 # Lean identifier characters (Lean's `isLetterLike` and `isSubScriptAlnum`), so
 # names such as `embedComplₗ` or `forall₂_le` are not truncated to an ASCII prefix.
 _LEAN_LETTER = (
@@ -67,12 +69,23 @@
     r"(?:_root_\.)?[A-Za-z_][A-Za-z0-9_'.]*"
     r"(?:\s+[A-Za-z_][A-Za-z0-9_'.]*)*\s*$"
 )
+CITATION_GROUP_RE = re.compile(
+    r"\[((?:\s*@[^\];,\s]+(?:\s*,\s*[^;\]\n]+)?\s*)"
+    r"(?:;\s*@[^\];,\s]+(?:\s*,\s*[^;\]\n]+)?\s*)*)\]"
+)
+SF_TARGET_RE = re.compile(r"(?:\]\(\s*|\]:\s*)sf:([^)\s]+)")
 
 
 class CatalogError(ValueError):
     """A deterministic catalog source or audit validation error."""
 
 
+@dataclass(frozen=True)
+class Citation:
+    key: str
+    at: str | None = None
+
+
 @dataclass(frozen=True)
 class CatalogItem:
     name: str
@@ -80,6 +93,7 @@ class CatalogItem:
     module: str | None
     label: str | None = None
     headline: bool = False
+    symcat: str | None = None
 
     @property
     def anchor(self) -> str:
@@ -95,6 +109,7 @@ class CatalogPage:
     authors: tuple[str, ...]
     years: tuple[int, ...]
     formalized: str
+    references: tuple[Citation, ...]
     definitions: tuple[CatalogItem, ...]
     theorems: tuple[CatalogItem, ...]
     content: str
@@ -108,6 +123,17 @@ def items(self) -> tuple[CatalogItem, ...]:
     def url(self) -> str:
         return f"{BASE_PATH}{self.section}/{self.slug}/"
 
+    @property
+    def citations(self) -> tuple[Citation, ...]:
+        """All listed and inline citations, in stable first-occurrence order."""
+        result: list[Citation] = []
+        seen: set[str] = set()
+        for citation in self.references + extract_citations(self.content):
+            if citation.key not in seen:
+                seen.add(citation.key)
+                result.append(citation)
+        return tuple(result)
+
 
 @dataclass(frozen=True)
 class SourceDeclaration:
@@ -238,12 +264,15 @@ def _read_items(path: str, raw: Any, expected_kind: str) -> tuple[CatalogItem, .
     result: list[CatalogItem] = []
     seen: set[str] = set()
     for index, record in enumerate(raw, start=1):
-        if not isinstance(record, dict) or set(record) - {"name", "module", "label", "headline"}:
+        if not isinstance(record, dict) or set(record) - {
+            "name", "module", "label", "headline", "symcat"
+        }:
             raise _source_error(path, f"invalid {expected_kind} record {index}")
         name = record.get("name")
         module = record.get("module")
         label = record.get("label")
         headline = record.get("headline", False)
+        symcat = record.get("symcat")
         if label is not None and (
             not isinstance(label, str) or not label.strip() or label.strip() != label or "\n" in label
         ):
@@ -254,6 +283,13 @@ def _read_items(path: str, raw: Any, expected_kind: str) -> tuple[CatalogItem, .
             raise _source_error(path, f"{expected_kind} record {index}: only theorems can be headlines")
         if headline and label is None:
             raise _source_error(path, f"{expected_kind} record {index}: a headline needs a label")
+        if symcat is not None and (
+            not isinstance(symcat, str)
+            or not symcat
+            or symcat.strip() != symcat
+            or any(character.isspace() for character in symcat)
+        ):
+            raise _source_error(path, f"{expected_kind} record {index} has an invalid symcat label")
         if not isinstance(name, str) or not NAME_RE.fullmatch(name):
             raise _source_error(path, f"{expected_kind} record {index} needs a fully qualified name")
         if module is not None:
@@ -272,11 +308,57 @@ def _read_items(path: str, raw: Any, expected_kind: str) -> tuple[CatalogItem, .
         if name in seen:
             raise _source_error(path, f"duplicate {expected_kind} declaration {name}")
         seen.add(name)
-        result.append(CatalogItem(name, expected_kind, module, label, headline))
+        result.append(CatalogItem(name, expected_kind, module, label, headline, symcat))
+    return tuple(result)
+
+
+def extract_citations(content: str) -> tuple[Citation, ...]:
+    """Extract the supported Pandoc-style citation groups from catalog Markdown."""
+    citations: list[Citation] = []
+    for group in CITATION_GROUP_RE.finditer(content):
+        for raw in group.group(1).split(";"):
+            key_and_at = raw.strip().removeprefix("@").split(",", 1)
+            key = key_and_at[0].strip()
+            at = key_and_at[1].strip() if len(key_and_at) == 2 else None
+            citations.append(Citation(key, at or None))
+    return tuple(citations)
+
+
+def extract_sf_labels(content: str) -> tuple[str, ...]:
+    """Return labels used as ``sf:`` Markdown link targets."""
+    return tuple(match.group(1) for match in SF_TARGET_RE.finditer(content))
+
+
+def _read_references(path: str, raw: Any) -> tuple[Citation, ...]:
+    if raw is None:
+        return ()
+    if not isinstance(raw, list):
+        raise _source_error(path, "references must be an array")
+    result: list[Citation] = []
+    seen: set[str] = set()
+    for index, item in enumerate(raw, start=1):
+        if isinstance(item, str):
+            key, at = item, None
+        elif isinstance(item, dict) and set(item) <= {"key", "at"}:
+            key, at = item.get("key"), item.get("at")
+        else:
+            raise _source_error(path, f"invalid reference record {index}")
+        if not isinstance(key, str) or not key or key.strip() != key or any(
+            character.isspace() for character in key
+        ):
+            raise _source_error(path, f"reference record {index} has an invalid key")
+        if at is not None and (
+            not isinstance(at, str) or not at or at.strip() != at or "\n" in at
+        ):
+            raise _source_error(path, f"reference record {index} has an invalid locator")
+        if key in seen:
+            raise _source_error(path, f"duplicate reference key {key}")
+        seen.add(key)
+        result.append(Citation(key, at))
     return tuple(result)
 
 
-def _published_title(path: str, content: str) -> str:
+def _published_title(path: str, content: str, has_reference_keys: bool = False) -> str:
     lines = content.splitlines()
     headings = [line[2:].strip() for line in lines if re.fullmatch(r"#\s+.+", line)]
     if len(headings) != 1:
@@ -287,7 +369,17 @@ def _published_title(path: str, content: str) -> str:
     )
     if reference_at is None:
         raise _source_error(path, 'published content must contain a "## References" section')
-    if not any(line.strip() for line in lines[reference_at + 1 :]):
+    reference_end = next(
+        (
+            index
+            for index, line in enumerate(lines[reference_at + 1 :], start=reference_at + 1)
+            if re.fullmatch(r"#{1,2}\s+.+", line)
+        ),
+        len(lines),
+    )
+    if not has_reference_keys and not any(
+        line.strip() for line in lines[reference_at + 1 : reference_end]
+    ):
         raise _source_error(path, "References must not be empty")
     return headings[0]
 
@@ -369,6 +461,7 @@ def parse_challenge_source(
         "authors",
         "years",
         "formalized",
+        "references",
         "definitions",
         "theorems",
     }
@@ -385,6 +478,7 @@ def parse_challenge_source(
         raise _source_error(source_path, "slug must be lowercase hyphenated text")
     authors, years = _read_attribution(source_path, metadata)
     formalized = _read_formalized(source_path, metadata, commit_date)
+    references = _read_references(source_path, metadata.get("references"))
     definitions = _read_items(source_path, metadata.get("definitions"), "definition")
     theorems = _read_items(source_path, metadata.get("theorems"), "theorem")
     if not definitions and not theorems:
@@ -397,10 +491,13 @@ def parse_challenge_source(
         authors=authors,
         years=years,
         formalized=formalized,
+        references=references,
         definitions=definitions,
         theorems=theorems,
         content=content,
-        title=_published_title(source_path, content),
+        title=_published_title(
+            source_path, content, bool(references or extract_citations(content))
+        ),
     )
 
 
@@ -902,6 +999,51 @@ def validate_audit_report(
     return report
 
 
+def load_symcat_snapshots(
+    repo_root: pathlib.Path,
+) -> tuple[dict[str, str], dict[str, dict[str, Any]]]:
+    """Load and validate the committed symmetricfunctions.com snapshots."""
+    directory = repo_root / "website" / "symcat"
+    labels_path = directory / "labels.json"
+    bibliography_path = directory / "bibliography.json"
+    try:
+        labels = json.loads(labels_path.read_text(encoding="utf-8"))
+        bibliography = json.loads(bibliography_path.read_text(encoding="utf-8"))
+    except (OSError, json.JSONDecodeError) as error:
+        raise CatalogError(f"cannot load SymCat snapshots: {error}") from error
+    if not isinstance(labels, dict) or not all(
+        isinstance(label, str)
+        and label
+        and isinstance(href, str)
+        and href
+        and not urlsplit(href).scheme
+        and not urlsplit(href).netloc
+        and not href.startswith("/")
+        for label, href in labels.items()
+    ):
+        raise CatalogError(f"{labels_path}: expected a mapping of labels to relative links")
+    if (
+        not isinstance(bibliography, dict)
+        or bibliography.get("schema_version") != 1
+        or set(bibliography) != {"schema_version", "entries"}
+        or not isinstance(bibliography["entries"], dict)
+    ):
+        raise CatalogError(f"{bibliography_path}: invalid bibliography snapshot schema")
+    entries = bibliography["entries"]
+    for key, entry in entries.items():
+        if (
+            not isinstance(key, str)
+            or not isinstance(entry, dict)
+            or set(entry) != {"label", "csl", "bibtex"}
+            or not isinstance(entry["label"], str)
+            or not entry["label"]
+            or not isinstance(entry["csl"], dict)
+            or not isinstance(entry["bibtex"], str)
+        ):
+            raise CatalogError(f"{bibliography_path}: invalid entry {key!r}")
+    return labels, entries
+
+
 def _safe_url(url: str) -> str | None:
     parsed = urlsplit(url)
     if parsed.scheme and parsed.scheme.lower() not in {"http", "https", "mailto"}:
@@ -915,7 +1057,14 @@ def _safe_url(url: str) -> str | None:
 LIST_ITEM_RE = re.compile(r"(\s*)(?:([-*])|(\d+)\.)\s+(.+)")
 
 
-def render_markdown(markdown: str) -> str:
+def render_markdown(
+    markdown: str,
+    *,
+    symcat_labels: dict[str, str] | None = None,
+    bibliography: dict[str, dict[str, Any]] | None = None,
+    base_path: str = BASE_PATH,
+    generated_references: str = "",
+) -> str:
     """Render the deliberate Markdown subset, escaping raw HTML by default.
 
     Supported: headings, paragraphs, `-`/`*` and numbered lists whose items may
@@ -944,7 +1093,7 @@ def render_markdown(markdown: str) -> str:
     code_indent = 0
 
     def inline(text: str) -> str:
-        return _inline(text, references)
+        return _inline(text, references, symcat_labels, bibliography, base_path)
 
     def flush_paragraph() -> None:
         if paragraph:
@@ -1009,7 +1158,22 @@ def flush_list() -> None:
         raise CatalogError("unterminated Markdown code fence")
     flush_paragraph()
     flush_list()
-    return "\n".join(blocks)
+    rendered = "\n".join(blocks)
+    if generated_references:
+        marker = '<h2 id="references">References</h2>'
+        marker_at = rendered.find(marker)
+        if marker_at == -1:
+            raise CatalogError('generated references need a "## References" section')
+        following = rendered.find("<h2 ", marker_at + len(marker))
+        insert_at = len(rendered) if following == -1 else following
+        rendered = (
+            rendered[:insert_at].rstrip()
+            + "\n"
+            + generated_references
+            + "\n"
+            + rendered[insert_at:]
+        )
+    return rendered
 
 
 def _anchor(text: str) -> str:
@@ -1017,8 +1181,16 @@ def _anchor(text: str) -> str:
     return plain or "section"
 
 
-def _inline(text: str, references: dict[str, str] | None = None) -> str:
+def _inline(
+    text: str,
+    references: dict[str, str] | None = None,
+    symcat_labels: dict[str, str] | None = None,
+    bibliography: dict[str, dict[str, Any]] | None = None,
+    base_path: str = BASE_PATH,
+) -> str:
     references = references or {}
+    symcat_labels = symcat_labels or {}
+    bibliography = bibliography or {}
     escaped = html.escape(text, quote=False)
     # protect code spans from emphasis and link rewriting
     codes: list[str] = []
@@ -1037,8 +1209,33 @@ def stash_math(match: re.Match[str]) -> str:
 
     rendered = re.sub(r"\$\$.+?\$\$|\$[^$\n]+?\$", stash_math, rendered)
 
+    def citation_group(match: re.Match[str]) -> str:
+        rendered_citations: list[str] = []
+        for raw in match.group(1).split(";"):
+            key_and_at = raw.strip().removeprefix("@").split(",", 1)
+            key = html.unescape(key_and_at[0].strip())
+            at = key_and_at[1].strip() if len(key_and_at) == 2 else None
+            entry = bibliography.get(key)
+            if entry is None:
+                return match.group(0)
+            label = html.escape(entry["label"])
+            suffix = f", {at}" if at else ""
+            rendered_citations.append(
+                f'<a class="citation" href="#ref-{html.escape(key, quote=True)}">'
+                f"[{label}{suffix}]</a>"
+            )
+        return "; ".join(rendered_citations)
+
+    rendered = CITATION_GROUP_RE.sub(citation_group, rendered)
+
     def anchor(label: str, target: str) -> str:
-        safe = _safe_url(html.unescape(target.strip()))
+        raw_target = html.unescape(target.strip())
+        if raw_target.startswith("sf:"):
+            href = symcat_labels.get(raw_target.removeprefix("sf:"))
+            raw_target = SYMCAT_ORIGIN + href if href else raw_target
+        elif raw_target.startswith(BASE_PATH) and base_path != BASE_PATH:
+            raw_target = base_path + raw_target.removeprefix(BASE_PATH)
+        safe = _safe_url(raw_target)
         if safe is None:
             return label
         return f'<a href="{html.escape(safe, quote=True)}">{label}</a>'
@@ -1057,6 +1254,111 @@ def reference(match: re.Match[str]) -> str:
     return re.sub(r"\x00(\d+)\x00", lambda m: f"<code>{codes[int(m.group(1))]}</code>", rendered)
 
 
+def _csl_authors(csl: dict[str, Any]) -> str:
+    authors = csl.get("author", [])
+    if not isinstance(authors, list):
+        return ""
+    names: list[str] = []
+    for author in authors:
+        if not isinstance(author, dict):
+            continue
+        literal = author.get("literal")
+        if isinstance(literal, str) and literal:
+            names.append(literal)
+            continue
+        given = author.get("given") if isinstance(author.get("given"), str) else ""
+        family = author.get("family") if isinstance(author.get("family"), str) else ""
+        name = " ".join(part for part in (given, family) if part)
+        if name:
+            names.append(name)
+    if len(names) == 2:
+        return " & ".join(names)
+    if len(names) > 2:
+        return ", ".join(names[:-1]) + ", & " + names[-1]
+    return "".join(names)
+
+
+def _csl_year(csl: dict[str, Any]) -> str:
+    issued = csl.get("issued")
+    if not isinstance(issued, dict):
+        return ""
+    parts = issued.get("date-parts")
+    if (
+        isinstance(parts, list)
+        and parts
+        and isinstance(parts[0], list)
+        and parts[0]
+        and isinstance(parts[0][0], int)
+    ):
+        return str(parts[0][0])
+    return ""
+
+
+def _reference_list(
+    citations: tuple[Citation, ...], bibliography: dict[str, dict[str, Any]]
+) -> str:
+    rows: list[str] = []
+    for citation in citations:
+        entry = bibliography[citation.key]
+        csl = entry["csl"]
+        authors = _csl_authors(csl)
+        title = csl.get("title") if isinstance(csl.get("title"), str) else citation.key
+        doi = csl.get("DOI") if isinstance(csl.get("DOI"), str) else ""
+        url = csl.get("URL") if isinstance(csl.get("URL"), str) else ""
+        target = f"https://doi.org/{doi}" if doi else url
+        title_html = html.escape(title)
+        if target and _safe_url(target):
+            title_html = f'<a href="{html.escape(target, quote=True)}">{title_html}</a>'
+        venue = csl.get("container-title", "")
+        if isinstance(venue, list):
+            venue = ", ".join(str(part) for part in venue)
+        if not isinstance(venue, str):
+            venue = ""
+        volume = csl.get("volume") if isinstance(csl.get("volume"), str) else ""
+        issue = csl.get("issue") if isinstance(csl.get("issue"), str) else ""
+        pages = csl.get("page") if isinstance(csl.get("page"), str) else ""
+        venue_parts = [venue]
+        if volume:
+            venue_parts.append(volume + (f"({issue})" if issue else ""))
+        elif issue:
+            venue_parts.append(f"({issue})")
+        if pages:
+            venue_parts.append(pages)
+        year = _csl_year(csl)
+        if year:
+            venue_parts.append(year)
+        details = ", ".join(part for part in venue_parts if part)
+        label = entry["label"] + (f", {citation.at}" if citation.at else "")
+        rows.append(
+            f'<li id="ref-{html.escape(citation.key, quote=True)}">'
+            f'<span class="reference-label">[{html.escape(label)}]</span> '
+            + (
+                f'<span class="reference-authors">{html.escape(authors)}.</span> '
+                if authors
+                else ""
+            )
+            + f'<span class="reference-title">{title_html}.</span> '
+            + (f'<span class="reference-venue">{html.escape(details)}.</span>' if details else "")
+            + '<details class="bibtex"><summary>BibTeX</summary><pre><code>'
+            + html.escape(entry["bibtex"].strip())
+            + "</code></pre></details></li>"
+        )
+    return '<ol class="reference-list">' + "".join(rows) + "</ol>"
+
+
+def _normalize_base_path(base_path: str) -> str:
+    if not isinstance(base_path, str) or not base_path.startswith("/"):
+        raise CatalogError("site base path must start with /")
+    if "?" in base_path or "#" in base_path or any(character.isspace() for character in base_path):
+        raise CatalogError("site base path must not contain whitespace, a query, or a fragment")
+    normalized = "/" + base_path.strip("/")
+    return "/" if normalized == "/" else normalized + "/"
+
+
+def _page_url(page: CatalogPage, base_path: str) -> str:
+    return f"{base_path}{page.section}/{page.slug}/"
+
+
 def _source_link(revision: str, source: SourceDeclaration) -> str:
     return (
         "https://github.com/PerAlexandersson/RealRooted/blob/"
@@ -1064,11 +1366,11 @@ def _source_link(revision: str, source: SourceDeclaration) -> str:
     )
 
 
-def _relative_url(target: str, current_url: str) -> str:
-    if not target.startswith(BASE_PATH) or not current_url.startswith(BASE_PATH):
+def _relative_url(target: str, current_url: str, base_path: str = BASE_PATH) -> str:
+    if not target.startswith(base_path) or not current_url.startswith(base_path):
         raise CatalogError("generator received a URL outside its base path")
-    current_relative = current_url.removeprefix(BASE_PATH).strip("/")
-    target_relative = target.removeprefix(BASE_PATH).strip("/")
+    current_relative = current_url.removeprefix(base_path).strip("/")
+    target_relative = target.removeprefix(base_path).strip("/")
     current_directory = pathlib.PurePosixPath(current_relative).parts if current_relative else ()
     target_parts = pathlib.PurePosixPath(target_relative).parts if target_relative else ()
     common = 0
@@ -1084,14 +1386,20 @@ def _relative_url(target: str, current_url: str) -> str:
     return relative
 
 
-def _template(repo_root: pathlib.Path, body: str, title: str, current_url: str) -> str:
+def _template(
+    repo_root: pathlib.Path,
+    body: str,
+    title: str,
+    current_url: str,
+    base_path: str = BASE_PATH,
+) -> str:
     template_path = repo_root / "website" / "templates" / "page.html"
     template = template_path.read_text(encoding="utf-8")
-    home = _relative_url(BASE_PATH, current_url)
-    stylesheet = _relative_url(f"{BASE_PATH}assets/site.css", current_url)
-    script = _relative_url(f"{BASE_PATH}assets/site.js", current_url)
-    results = _relative_url(f"{BASE_PATH}results/", current_url)
-    authors = _relative_url(f"{BASE_PATH}authors/", current_url)
+    home = _relative_url(base_path, current_url, base_path)
+    stylesheet = _relative_url(f"{base_path}assets/site.css", current_url, base_path)
+    script = _relative_url(f"{base_path}assets/site.js", current_url, base_path)
+    results = _relative_url(f"{base_path}results/", current_url, base_path)
+    authors = _relative_url(f"{base_path}authors/", current_url, base_path)
     return (
         template.replace("{{ title }}", html.escape(title))
         .replace("{{ body }}", body)
@@ -1228,7 +1536,10 @@ def _catalog_group(identifier: str, title: str, intro: str, rows: str) -> str:
 
 
 def _item_list(
-    items: Iterable[CatalogItem], resolved: dict[str, SourceDeclaration], revision: str
+    items: Iterable[CatalogItem],
+    resolved: dict[str, SourceDeclaration],
+    revision: str,
+    symcat_labels: dict[str, str],
 ) -> str:
     rows: list[str] = []
     for item in items:
@@ -1241,6 +1552,14 @@ def _item_list(
                 + ("<span class=\"headline-tag\">Main result</span>" if item.headline else "")
                 + "</p>"
             )
+        symcat_link = ""
+        if item.symcat:
+            symcat_url = SYMCAT_ORIGIN + symcat_labels[item.symcat]
+            symcat_link = (
+                '<a class="symcat-source" href="'
+                + html.escape(symcat_url, quote=True)
+                + '">statement on symmetricfunctions.com <span aria-hidden="true">↗</span></a>'
+            )
         rows.append(
             f"<li id=\"{item.anchor}\">"
             + label
@@ -1248,17 +1567,21 @@ def _item_list(
             + html.escape(item.name)
             + "</code><a class=\"source\" href=\""
             + html.escape(_source_link(revision, source), quote=True)
-            + "\">source</a></div><pre class=\"lean-declaration\"><code>"
+            + '">source</a>'
+            + symcat_link
+            + '</div><pre class="lean-declaration"><code>'
             + html.escape(source.source_code)
             + "</code></pre></li>"
         )
     return "<ul class=\"declarations\">" + "".join(rows) + "</ul>"
 
 
-def _authors_page(entries: Iterable[AuthorIndexEntry]) -> str:
+def _authors_page(
+    entries: Iterable[AuthorIndexEntry], base_path: str = BASE_PATH
+) -> str:
     """Render the page-level credits index without adding editorial data to Lean files."""
     entries = tuple(entries)
-    current_url = f"{BASE_PATH}authors/"
+    current_url = f"{base_path}authors/"
     rows: list[str] = []
     for entry in entries:
         heading = html.escape(entry.display_name)
@@ -1270,7 +1593,7 @@ def _authors_page(entries: Iterable[AuthorIndexEntry]) -> str:
         credits: list[str] = []
         for page in entry.pages:
             years = _years_label(page.years) or "year not recorded"
-            href = _relative_url(page.url, current_url)
+            href = _relative_url(_page_url(page, base_path), current_url, base_path)
             credits.append(
                 '<li><a class="author-credit-title" href="'
                 + html.escape(href, quote=True)
@@ -1370,7 +1693,7 @@ def handle_starttag(self, tag: str, attrs: list[tuple[str, str | None]]) -> None
             self.hrefs.extend(value for name, value in attrs if name == "href" and value is not None)
 
 
-def validate_internal_links(files: dict[str, str]) -> None:
+def validate_internal_links(files: dict[str, str], base_path: str = BASE_PATH) -> None:
     """Reject generated links that cannot resolve under the project-site base path."""
     known = set(files)
     for source_name, document in files.items():
@@ -1382,9 +1705,9 @@ def validate_internal_links(files: dict[str, str]) -> None:
             if parsed.scheme or parsed.netloc or href.startswith("#"):
                 continue
             if href.startswith("/"):
-                if not href.startswith(BASE_PATH):
-                    raise CatalogError(f"{source_name}: link escapes {BASE_PATH}: {href}")
-                target_path = href[len(BASE_PATH) :].split("#", 1)[0]
+                if not href.startswith(base_path):
+                    raise CatalogError(f"{source_name}: link escapes {base_path}: {href}")
+                target_path = href[len(base_path) :].split("#", 1)[0]
                 target = target_path
             else:
                 target_path = href.split("#", 1)[0]
@@ -1402,8 +1725,32 @@ def render_site(
     pages: tuple[CatalogPage, ...],
     resolved: dict[str, SourceDeclaration],
     revision: str,
+    *,
+    base_path: str = BASE_PATH,
+    site_origin: str = SITE_ORIGIN,
 ) -> dict[str, str]:
     """Render deterministic generated files keyed by their output-relative path."""
+    base_path = _normalize_base_path(base_path)
+    parsed_origin = urlsplit(site_origin)
+    if parsed_origin.scheme not in {"http", "https"} or not parsed_origin.netloc:
+        raise CatalogError("site origin must be an absolute HTTP(S) URL")
+    site_origin = site_origin.rstrip("/")
+    symcat_labels, bibliography = load_symcat_snapshots(repo_root)
+    for page in pages:
+        for citation in page.citations:
+            if citation.key not in bibliography:
+                raise _source_error(page.source_path, f"unknown bibliography key {citation.key}")
+        for label in extract_sf_labels(page.content):
+            if label not in symcat_labels:
+                raise _source_error(
+                    page.source_path, f"unknown symmetricfunctions.com label {label}"
+                )
+        for item in page.items:
+            if item.symcat and item.symcat not in symcat_labels:
+                raise _source_error(
+                    page.source_path,
+                    f"unknown symmetricfunctions.com label {item.symcat} for {item.name}",
+                )
     files: dict[str, str] = {}
     author_entries = build_author_index(pages, load_author_profiles(repo_root))
     grouped = {section: [page for page in pages if page.section == section] for section in SECTIONS}
@@ -1460,7 +1807,9 @@ def render_site(
         )
         + "</div></main>"
     )
-    files["index.html"] = _template(repo_root, index_body, "RealRooted catalog", BASE_PATH)
+    files["index.html"] = _template(
+        repo_root, index_body, "RealRooted catalog", base_path, base_path
+    )
     for section, section_pages in grouped.items():
         links = "".join(_page_card(page, f"{page.slug}/") for page in section_pages)
         body = (
@@ -1469,11 +1818,21 @@ def render_site(
             + _sort_controls()
             + f'<ul class="section-index" data-catalog-list>{links}</ul></main>'
         )
-        section_url = f"{BASE_PATH}{section}/"
-        files[f"{section}/index.html"] = _template(repo_root, body, section.title(), section_url)
+        section_url = f"{base_path}{section}/"
+        files[f"{section}/index.html"] = _template(
+            repo_root, body, section.title(), section_url, base_path
+        )
     for page in pages:
-        definition_html = _item_list(page.definitions, resolved, revision) if page.definitions else ""
-        theorem_html = _item_list(page.theorems, resolved, revision) if page.theorems else ""
+        definition_html = (
+            _item_list(page.definitions, resolved, revision, symcat_labels)
+            if page.definitions
+            else ""
+        )
+        theorem_html = (
+            _item_list(page.theorems, resolved, revision, symcat_labels)
+            if page.theorems
+            else ""
+        )
         selected = ""
         if definition_html:
             selected += (
@@ -1495,7 +1854,15 @@ def render_site(
             + html.escape(page.section.title())
             + "</a></p>"
             + "<article class=\"prose\">"
-            + render_markdown(page.content)
+            + render_markdown(
+                page.content,
+                symcat_labels=symcat_labels,
+                bibliography=bibliography,
+                base_path=base_path,
+                generated_references=_reference_list(page.citations, bibliography)
+                if page.citations
+                else "",
+            )
             + "</article><div class=\"lean-results\">"
             + selected
             + "</div>"
@@ -1505,13 +1872,21 @@ def render_site(
             + f"Formalized {page.formalized}</span>.</p></main>"
         )
         files[f"{page.section}/{page.slug}/index.html"] = _template(
-            repo_root, body, page.title, page.url
+            repo_root, body, page.title, _page_url(page, base_path), base_path
         )
     files["results/index.html"] = _template(
-        repo_root, _results_page(pages, resolved, revision), "All results", f"{BASE_PATH}results/"
+        repo_root,
+        _results_page(pages, resolved, revision),
+        "All results",
+        f"{base_path}results/",
+        base_path,
     )
     files["authors/index.html"] = _template(
-        repo_root, _authors_page(author_entries), "Authors", f"{BASE_PATH}authors/"
+        repo_root,
+        _authors_page(author_entries, base_path),
+        "Authors",
+        f"{base_path}authors/",
+        base_path,
     )
     manifest = {
         "schema_version": 1,
@@ -1520,7 +1895,7 @@ def render_site(
         "pages": [
             {
                 "title": page.title,
-                "url": page.url,
+                "url": _page_url(page, base_path),
                 "source_path": page.source_path,
                 "authors": list(page.authors),
                 "years": list(page.years),
@@ -1529,6 +1904,15 @@ def render_site(
                 "theorems": [item.name for item in page.theorems],
                 "headlines": [item.name for item in page.theorems if item.headline],
                 "labels": {item.name: item.label for item in page.items if item.label},
+                "declarations": [
+                    {
+                        "name": item.name,
+                        "kind": item.expected_kind,
+                        "url": site_origin + _page_url(page, base_path) + f"#{item.anchor}",
+                        **({"symcat": item.symcat} if item.symcat else {}),
+                    }
+                    for item in page.items
+                ],
             }
             for page in pages
         ],
@@ -1538,7 +1922,7 @@ def render_site(
     files["assets/site.css"] = asset.read_text(encoding="utf-8")
     script = repo_root / "website" / "assets" / "site.js"
     files["assets/site.js"] = script.read_text(encoding="utf-8")
-    validate_internal_links(files)
+    validate_internal_links(files, base_path)
     return files
 
 
diff --git a/scripts/symcat_sync.py b/scripts/symcat_sync.py
new file mode 100644
index 000000000..5448e17c6
--- /dev/null
+++ b/scripts/symcat_sync.py
@@ -0,0 +1,178 @@
+#!/usr/bin/env python3
+"""Refresh the catalog's symmetricfunctions.com label and bibliography snapshots."""
+
+from __future__ import annotations
+
+import argparse
+import json
+import pathlib
+import sys
+from typing import Any
+from urllib.parse import urljoin, urlsplit
+from urllib.request import urlopen
+
+sys.dont_write_bytecode = True
+
+from challenge_catalog import CatalogError, extract_sf_labels, load_catalog
+
+
+LIVE_ROOT = "https://www.symmetricfunctions.com/"
+CSL_FIELDS = (
+    "author",
+    "title",
+    "container-title",
+    "volume",
+    "issue",
+    "page",
+    "issued",
+    "DOI",
+    "URL",
+    "type",
+)
+
+
+def _source_at(root: str, filename: str) -> str:
+    parsed = urlsplit(root)
+    if parsed.scheme in {"http", "https"}:
+        return urljoin(root.rstrip("/") + "/", filename)
+    return str(pathlib.Path(root) / filename)
+
+
+def _read_json(source: str) -> Any:
+    parsed = urlsplit(source)
+    try:
+        if parsed.scheme in {"http", "https"}:
+            with urlopen(source, timeout=60) as response:
+                return json.load(response)
+        return json.loads(pathlib.Path(source).read_text(encoding="utf-8"))
+    except (OSError, ValueError) as error:
+        raise CatalogError(f"cannot read {source}: {error}") from error
+
+
+def build_snapshots(
+    repo_root: pathlib.Path,
+    labels_upstream: Any,
+    bibliography_upstream: Any,
+    keep: tuple[str, ...] = (),
+) -> tuple[dict[str, str], dict[str, Any]]:
+    """Select and normalize the upstream data needed by the current catalog."""
+    if not isinstance(labels_upstream, dict):
+        raise CatalogError("upstream site-labels.json must contain an object")
+    if (
+        not isinstance(bibliography_upstream, dict)
+        or bibliography_upstream.get("schema_version") != 1
+        or not isinstance(bibliography_upstream.get("entries"), dict)
+    ):
+        raise CatalogError("upstream bibliography.json has an invalid schema")
+
+    pages = load_catalog(repo_root)
+    used_labels = sorted(
+        {
+            label
+            for page in pages
+            for label in (
+                *(item.symcat for item in page.items if item.symcat),
+                *extract_sf_labels(page.content),
+            )
+        }
+    )
+    cited_keys = sorted({citation.key for page in pages for citation in page.citations} | set(keep))
+
+    missing_labels = [label for label in used_labels if label not in labels_upstream]
+    if missing_labels:
+        raise CatalogError("upstream labels missing: " + ", ".join(missing_labels))
+    upstream_entries = bibliography_upstream["entries"]
+    missing_keys = [key for key in cited_keys if key not in upstream_entries]
+    if missing_keys:
+        raise CatalogError("upstream bibliography missing: " + ", ".join(missing_keys))
+
+    labels: dict[str, str] = {}
+    for label in sorted(labels_upstream):
+        record = labels_upstream[label]
+        href = record.get("href") if isinstance(record, dict) else None
+        if not isinstance(label, str) or not isinstance(href, str) or not href:
+            raise CatalogError(f"invalid upstream label record {label!r}")
+        labels[label] = href
+
+    selected: dict[str, dict[str, Any]] = {}
+    for key in cited_keys:
+        record = upstream_entries[key]
+        if not isinstance(record, dict):
+            raise CatalogError(f"invalid upstream bibliography entry {key!r}")
+        label = record.get("label")
+        csl = record.get("csl")
+        bibtex = record.get("bibtex")
+        if (
+            not isinstance(label, str)
+            or not label
+            or not isinstance(csl, dict)
+            or not isinstance(bibtex, str)
+        ):
+            raise CatalogError(f"invalid upstream bibliography entry {key!r}")
+        selected[key] = {
+            "label": label,
+            "csl": {field: csl[field] for field in CSL_FIELDS if field in csl},
+            "bibtex": bibtex.strip() + "\n",
+        }
+    return labels, {"schema_version": 1, "entries": selected}
+
+
+def write_snapshots(
+    repo_root: pathlib.Path, labels: dict[str, str], bibliography: dict[str, Any]
+) -> None:
+    target = repo_root / "website" / "symcat"
+    target.mkdir(parents=True, exist_ok=True)
+    (target / "labels.json").write_text(
+        json.dumps(labels, ensure_ascii=False, sort_keys=True, separators=(",", ":")) + "\n",
+        encoding="utf-8",
+    )
+    (target / "bibliography.json").write_text(
+        json.dumps(bibliography, ensure_ascii=False, indent=2, sort_keys=True) + "\n",
+        encoding="utf-8",
+    )
+
+
+def parse_args() -> argparse.Namespace:
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument(
+        "source",
+        nargs="?",
+        default=LIVE_ROOT,
+        help="directory or URL containing site-labels.json and bibliography.json",
+    )
+    parser.add_argument(
+        "--repo-root",
+        type=pathlib.Path,
+        default=pathlib.Path(__file__).resolve().parents[1],
+    )
+    parser.add_argument("--labels-source", help="override the labels JSON path or URL")
+    parser.add_argument("--bibliography-source", help="override the bibliography JSON path or URL")
+    parser.add_argument("--keep", action="append", default=[], metavar="KEY")
+    return parser.parse_args()
+
+
+def main() -> int:
+    args = parse_args()
+    labels_source = args.labels_source or _source_at(args.source, "site-labels.json")
+    bibliography_source = args.bibliography_source or _source_at(
+        args.source, "bibliography.json"
+    )
+    try:
+        labels, bibliography = build_snapshots(
+            args.repo_root.resolve(),
+            _read_json(labels_source),
+            _read_json(bibliography_source),
+            tuple(args.keep),
+        )
+        write_snapshots(args.repo_root.resolve(), labels, bibliography)
+        print(
+            f"ok: wrote {len(labels)} labels and {len(bibliography['entries'])} bibliography entries"
+        )
+        return 0
+    except CatalogError as error:
+        print(f"error: {error}", file=sys.stderr)
+        return 1
+
+
+if __name__ == "__main__":
+    raise SystemExit(main())
diff --git a/scripts/test_challenge_catalog.py b/scripts/test_challenge_catalog.py
index fd2a67a65..c5a282be8 100644
--- a/scripts/test_challenge_catalog.py
+++ b/scripts/test_challenge_catalog.py
@@ -38,6 +38,7 @@ def catalog_block(
     slug: str = "sample",
     authors: str = "",
     years: str = "",
+    references: str = "",
     formalized: str = 'formalized = "2026-01-02"',
     definitions: str = "",
     theorems: str = '[[theorems]]\nname = "RealRooted.Challenges.Sample.proven"',
@@ -52,6 +53,7 @@ def catalog_block(
 {formalized}
 {authors}
 {years}
+{references}
 {definitions}
 {theorems}
 -->
@@ -93,6 +95,22 @@ def sample_source(self, *, extra: str = "") -> str:
             + "end RealRooted.Challenges.Sample\n"
         )
 
+    def write_symcat(
+        self,
+        *,
+        labels: dict[str, str] | None = None,
+        entries: dict | None = None,
+    ) -> None:
+        directory = self.root / "website" / "symcat"
+        directory.mkdir(parents=True, exist_ok=True)
+        (directory / "labels.json").write_text(
+            json.dumps(labels or {}, sort_keys=True), encoding="utf-8"
+        )
+        (directory / "bibliography.json").write_text(
+            json.dumps({"schema_version": 1, "entries": entries or {}}, sort_keys=True),
+            encoding="utf-8",
+        )
+
     def pages_and_sources(self) -> tuple[tuple, dict]:
         self.write("RealRooted/Challenges/Sample.lean", self.sample_source())
         pages = load_catalog(self.root)
@@ -676,6 +694,152 @@ def test_labels_do_not_change_the_catalog_digest(self) -> None:
         )
         self.assertEqual(catalog_digest(load_catalog(self.root)), plain)
 
+    def test_symcat_citations_reference_list_and_manifest(self) -> None:
+        content = (
+            "# Sample\n\nSee [the statement](sf:sampleTheorem) "
+            "[@Brenti1995, Thm. 2.4.2; @Other].\n\n"
+            "## References\n\nA short contextual note."
+        )
+        theorems = (
+            '[[theorems]]\nname = "RealRooted.Challenges.Sample.proven"\n'
+            'label = "Proven"\nheadline = true\nsymcat = "sampleTheorem"'
+        )
+        self.write(
+            "RealRooted/Challenges/Sample.lean",
+            catalog_block(
+                references='references = [{key = "Other", at = "Cor. 3"}]',
+                theorems=theorems,
+                content=content,
+            )
+            + "namespace RealRooted.Challenges.Sample\n"
+            + "theorem proven : True := by trivial\n"
+            + "end RealRooted.Challenges.Sample\n",
+        )
+        self.write_symcat(
+            labels={"sampleTheorem": "realRooted.htm#sampleTheorem"},
+            entries={
+                "Brenti1995": {
+                    "label": "Bre95",
+                    "csl": {
+                        "author": [{"given": "Francesco", "family": "Brenti"}],
+                        "title": "A paper",
+                        "container-title": "A Journal",
+                        "volume": "4",
+                        "issue": "2",
+                        "page": "1-9",
+                        "issued": {"date-parts": [[1995]]},
+                        "DOI": "10.1000/example",
+                    },
+                    "bibtex": "@article{Brenti1995, title={A paper}}\n",
+                },
+                "Other": {
+                    "label": "Oth20",
+                    "csl": {"title": "Other paper", "issued": {"date-parts": [[2020]]}},
+                    "bibtex": "@misc{Other, title={Other paper}}\n",
+                },
+            },
+        )
+        pages = load_catalog(self.root)
+        self.assertEqual(pages[0].references[0].key, "Other")
+        self.assertEqual(pages[0].theorems[0].symcat, "sampleTheorem")
+        output = render_site(self.root, pages, validate_sources(self.root, pages), REVISION)
+        page = output["families/sample/index.html"]
+        self.assertIn('href="#ref-Brenti1995">[Bre95, Thm. 2.4.2]</a>', page)
+        self.assertIn('href="#ref-Other">[Oth20]</a>', page)
+        self.assertIn(
+            'href="https://www.symmetricfunctions.com/realRooted.htm#sampleTheorem"', page
+        )
+        self.assertIn("statement on symmetricfunctions.com", page)
+        self.assertIn('id="ref-Brenti1995"', page)
+        self.assertIn("Francesco Brenti", page)
+        self.assertIn("https://doi.org/10.1000/example", page)
+        self.assertIn("BibTeX", page)
+        self.assertGreater(page.index("A short contextual note."), page.index("References</h2>"))
+        self.assertGreater(
+            page.index('class="reference-list"'), page.index("A short contextual note.")
+        )
+        manifest = json.loads(output["catalog-manifest.json"])
+        declaration = manifest["pages"][0]["declarations"][0]
+        self.assertEqual(declaration["symcat"], "sampleTheorem")
+        self.assertEqual(
+            declaration["url"],
+            "https://peralexandersson.github.io/RealRooted/families/sample/"
+            "#decl-RealRooted-Challenges-Sample-proven",
+        )
+
+    def test_unknown_bibliography_keys_and_symcat_labels_fail(self) -> None:
+        self.write_symcat()
+        for content, theorem, message in (
+            (
+                "# Sample\n\nUnknown [@Missing].\n\n## References\n",
+                '[[theorems]]\nname = "RealRooted.Challenges.Sample.proven"',
+                "unknown bibliography key Missing",
+            ),
+            (
+                "# Sample\n\n[Unknown](sf:missing).\n\n## References\n\n- prose",
+                '[[theorems]]\nname = "RealRooted.Challenges.Sample.proven"',
+                "unknown symmetricfunctions.com label missing",
+            ),
+            (
+                "# Sample\n\nText.\n\n## References\n\n- prose",
+                '[[theorems]]\nname = "RealRooted.Challenges.Sample.proven"\n'
+                'symcat = "missing"',
+                "unknown symmetricfunctions.com label missing",
+            ),
+        ):
+            with self.subTest(message=message):
+                self.write(
+                    "RealRooted/Challenges/Sample.lean",
+                    catalog_block(theorems=theorem, content=content)
+                    + "namespace RealRooted.Challenges.Sample\n"
+                    + "theorem proven : True := by trivial\n"
+                    + "end RealRooted.Challenges.Sample\n",
+                )
+                pages = load_catalog(self.root)
+                with self.assertRaisesRegex(CatalogError, message):
+                    render_site(self.root, pages, validate_sources(self.root, pages), REVISION)
+
+    def test_configurable_base_path_rewrites_legacy_absolute_links(self) -> None:
+        content = (
+            "# Sample\n\nSee [another page](/RealRooted/families/sample/).\n\n"
+            "## References\n\n- prose"
+        )
+        self.write("RealRooted/Challenges/Sample.lean", self.sample_source().replace(
+            "# Sample\n\nA checked result.\n\n## References\n\n- [Paper](https://example.org/)",
+            content,
+        ))
+        pages = load_catalog(self.root)
+        output = render_site(
+            self.root,
+            pages,
+            validate_sources(self.root, pages),
+            REVISION,
+            base_path="/",
+            site_origin="https://lean.symmetricfunctions.com",
+        )
+        self.assertIn('href="/families/sample/"', output["families/sample/index.html"])
+        self.assertNotIn('href="/RealRooted/', output["families/sample/index.html"])
+        declaration = json.loads(output["catalog-manifest.json"])["pages"][0]["declarations"][0]
+        self.assertEqual(
+            declaration["url"],
+            "https://lean.symmetricfunctions.com/families/sample/"
+            "#decl-RealRooted-Challenges-Sample-family",
+        )
+
+    def test_reference_section_may_be_populated_only_by_metadata(self) -> None:
+        source = catalog_block(
+            references='references = ["Only"]',
+            content="# Sample\n\nText.\n\n## References\n",
+        )
+        page = parse_challenge_source("Sample.lean", source, date(2026, 1, 2))
+        self.assertEqual(page.references[0].key, "Only")
+        with self.assertRaisesRegex(CatalogError, "References must not be empty"):
+            parse_challenge_source(
+                "Sample.lean",
+                catalog_block(content="# Sample\n\nText.\n\n## References\n"),
+                date(2026, 1, 2),
+            )
+
     def test_attribution_metadata_is_validated(self) -> None:
         bad_authors = catalog_block(authors='authors = ["Repeated", "Repeated"]')
         self.write("RealRooted/Challenges/Sample.lean", bad_authors)
diff --git a/scripts/test_symcat_sync.py b/scripts/test_symcat_sync.py
new file mode 100644
index 000000000..6f70e2577
--- /dev/null
+++ b/scripts/test_symcat_sync.py
@@ -0,0 +1,117 @@
+#!/usr/bin/env python3
+"""Unit tests for the symmetricfunctions.com snapshot synchronizer."""
+
+from __future__ import annotations
+
+import json
+import pathlib
+import sys
+import tempfile
+import unittest
+
+sys.dont_write_bytecode = True
+
+from challenge_catalog import CatalogError
+from symcat_sync import build_snapshots, write_snapshots
+
+
+SOURCE = '''/-!
+<!-- realrooted-catalog
+version = 1
+section = "theorems"
+slug = "sample"
+formalized = "2026-01-02"
+references = ["Cited"]
+
+[[theorems]]
+name = "RealRooted.Challenges.Sample.proven"
+symcat = "statement"
+-->
+
+<!-- realrooted-catalog-content -->
+# Sample
+
+See [context](sf:context) and [@Inline].
+
+## References
+<!-- /realrooted-catalog-content -->
+-/
+namespace RealRooted.Challenges.Sample
+theorem proven : True := by trivial
+end RealRooted.Challenges.Sample
+'''
+
+
+def bibliography_entry(key: str) -> dict:
+    return {
+        "label": key[:3],
+        "csl": {
+            "id": key,
+            "author": [{"family": "Example", "given": "A."}],
+            "title": f"{key} title",
+            "issued": {"date-parts": [[2026]]},
+            "DOI": f"10.1000/{key}",
+            "unused": "drop me",
+        },
+        "bibtex": f"@misc{{{key}}}\n\n",
+    }
+
+
+class SymcatSyncTest(unittest.TestCase):
+    def setUp(self) -> None:
+        self.tempdir = tempfile.TemporaryDirectory()
+        self.root = pathlib.Path(self.tempdir.name)
+        path = self.root / "RealRooted" / "Challenges" / "Sample.lean"
+        path.parent.mkdir(parents=True)
+        path.write_text(SOURCE, encoding="utf-8")
+
+    def tearDown(self) -> None:
+        self.tempdir.cleanup()
+
+    def upstream(self) -> tuple[dict, dict]:
+        labels = {
+            "unused": {"href": "unused.htm#unused", "page": "unused", "title": "Unused"},
+            "context": {"href": "context.htm#context", "page": "context", "title": "Context"},
+            "statement": {
+                "href": "statement.htm#statement",
+                "page": "statement",
+                "title": "Statement",
+            },
+        }
+        entries = {
+            key: bibliography_entry(key) for key in ("Cited", "Inline", "Kept", "Unused")
+        }
+        return labels, {"schema_version": 1, "entries": entries}
+
+    def test_selects_citations_keeps_extras_and_trims_csl(self) -> None:
+        labels_upstream, bibliography_upstream = self.upstream()
+        labels, bibliography = build_snapshots(
+            self.root, labels_upstream, bibliography_upstream, ("Kept",)
+        )
+        self.assertEqual(list(labels), ["context", "statement", "unused"])
+        self.assertEqual(list(bibliography["entries"]), ["Cited", "Inline", "Kept"])
+        self.assertNotIn("id", bibliography["entries"]["Cited"]["csl"])
+        self.assertNotIn("unused", bibliography["entries"]["Cited"]["csl"])
+        self.assertTrue(bibliography["entries"]["Cited"]["bibtex"].endswith("\n"))
+
+        write_snapshots(self.root, labels, bibliography)
+        first = (self.root / "website" / "symcat" / "bibliography.json").read_text()
+        write_snapshots(self.root, labels, bibliography)
+        self.assertEqual(
+            first, (self.root / "website" / "symcat" / "bibliography.json").read_text()
+        )
+        self.assertEqual(json.loads(first), bibliography)
+
+    def test_missing_cited_key_or_used_label_is_an_error(self) -> None:
+        labels, bibliography = self.upstream()
+        del bibliography["entries"]["Inline"]
+        with self.assertRaisesRegex(CatalogError, "upstream bibliography missing: Inline"):
+            build_snapshots(self.root, labels, bibliography)
+        labels, bibliography = self.upstream()
+        del labels["context"]
+        with self.assertRaisesRegex(CatalogError, "upstream labels missing: context"):
+            build_snapshots(self.root, labels, bibliography)
+
+
+if __name__ == "__main__":
+    unittest.main()
diff --git a/website/README.md b/website/README.md
index 8e5fc9411..35aacc367 100644
--- a/website/README.md
+++ b/website/README.md
@@ -12,12 +12,14 @@ section = "families"
 slug = "example"
 authors = ["Example", "Author"]
 years = [1952]
+references = ["Example1952", {key = "Other1953", at = "Thm. 2"}]
 
 [[definitions]]
 name = "RealRooted.Challenges.Example.family"
 
 [[theorems]]
 name = "RealRooted.Challenges.Example.realRooted"
+symcat = "exampleRealRooted"
 -->
 
 <!-- realrooted-catalog-content -->
@@ -32,6 +34,21 @@ Short human-readable account of the exact checked results.
 -/
 ```
 
+Page-level `references` entries are bibliography keys exported by
+`symmetricfunctions.com`; a record may add an `at` locator. In prose, `[@Example1952]`,
+`[@Example1952, Thm. 2]`, and `[@Example1952; @Other1953]` produce linked short citations and
+a generated bibliography in the existing `## References` section. Links with an `sf:` target,
+such as `[related statement](sf:exampleRealRooted)`, resolve through the committed SymCat label
+snapshot. A declaration's optional `symcat` label adds a link beside its Lean source link and is
+exported in `catalog-manifest.json`.
+
+Refresh both snapshots from the live site, or from a local directory containing the two exports:
+
+```bash
+python3 scripts/symcat_sync.py
+python3 scripts/symcat_sync.py /path/to/exports --keep ExtraBibliographyKey
+```
+
 `version`, `section`, and `slug` are required.  Sections are `concepts`,
 `families`, and `theorems`; slugs are lowercase hyphenated text.  Optional
 `authors` and `years` arrays record compact attribution already supported by
@@ -121,6 +138,10 @@ Build a source-only local preview without invoking Lake or Lean:
 python3 scripts/build_challenge_pages.py --preview
 ```
 
+Use `--base-path /` when previewing the layout intended for a root-served custom domain. The
+default remains `/RealRooted/`; `--site-origin` controls absolute declaration URLs in the
+manifest. Legacy prose links beginning with `/RealRooted/` are rewritten to the selected base.
+
 The command writes to the system temporary directory and prints a ready-to-run
 `python3 -m http.server` command.  It does not perform the post-Lean publication
 audit; that remains a CI/release check rather than a prerequisite for styling
diff --git a/website/assets/site.css b/website/assets/site.css
index 1d7e4d4d1..3253da8be 100644
--- a/website/assets/site.css
+++ b/website/assets/site.css
@@ -425,11 +425,12 @@ h3 { font-size: 1.25rem; }
 .declaration-meta {
   display: flex;
   align-items: center;
-  justify-content: space-between;
+  justify-content: flex-start;
   gap: 0.8rem;
   margin-bottom: 0.65rem;
 }
 .declaration-name { color: var(--ink); font-weight: 700; }
+.declaration-name + .source { margin-left: auto; }
 .lean-declaration {
   margin: 0;
   padding: 0.72rem 0.85rem;
@@ -454,6 +455,13 @@ h3 { font-size: 1.25rem; }
   font-weight: 750;
   text-decoration: none;
 }
+.symcat-source {
+  flex: none;
+  color: var(--muted);
+  font-size: 0.68rem;
+  font-weight: 650;
+  text-decoration: none;
+}
 .declaration-group--definition .source {
   background: var(--definition-soft);
   color: var(--definition);
@@ -461,6 +469,25 @@ h3 { font-size: 1.25rem; }
 code, pre { font-family: "SFMono-Regular", Consolas, "Liberation Mono", monospace; }
 code { overflow-wrap: anywhere; font-size: 0.88em; }
 .prose code { padding: 0.12rem 0.35rem; border-radius: 0.32rem; background: var(--accent-soft); }
+.citation { white-space: nowrap; }
+.reference-list {
+  display: grid;
+  gap: 0.9rem;
+  margin: 1rem 0 0;
+  padding: 0;
+  list-style: none;
+}
+.reference-list > li { scroll-margin-top: 6rem; }
+.reference-label { color: var(--accent-strong); font-weight: 750; }
+.reference-title { font-style: italic; }
+.reference-venue { color: var(--muted); }
+.bibtex {
+  margin-top: 0.3rem;
+  color: var(--muted);
+  font-family: ui-sans-serif, system-ui, sans-serif;
+}
+.bibtex summary { width: max-content; cursor: pointer; font-size: 0.68rem; font-weight: 750; }
+.bibtex pre { margin: 0.35rem 0 0; font-size: 0.7rem; }
 pre {
   overflow-x: auto;
   padding: 1rem;
diff --git a/website/symcat/bibliography.json b/website/symcat/bibliography.json
new file mode 100644
index 000000000..4793ba669
--- /dev/null
+++ b/website/symcat/bibliography.json
@@ -0,0 +1,3519 @@
+{
+  "entries": {
+    "AissenSchoenbergWhitney1952": {
+      "bibtex": "@article{AissenSchoenbergWhitney1952,\n  author = {Aissen, Michael and Schoenberg, I. J. and Whitney, A. M.},\n  title = {On the generating functions of totally positive sequences {I}},\n  year = {1952},\n  journal = {Journal d'Analyse Mathématique},\n  volume = {2},\n  number = {1},\n  pages = {93--103},\n  doi = {10.1007/BF02786970},\n  url = {https://doi.org/10.1007/BF02786970}\n}\n",
+      "csl": {
+        "DOI": "10.1007/BF02786970",
+        "URL": "https://doi.org/10.1007/BF02786970",
+        "author": [
+          {
+            "family": "Aissen",
+            "given": "Michael"
+          },
+          {
+            "family": "Schoenberg",
+            "given": "I. J."
+          },
+          {
+            "family": "Whitney",
+            "given": "A. M."
+          }
+        ],
+        "container-title": "Journal d’Analyse Mathématique",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              1952
+            ]
+          ]
+        },
+        "page": "93-103",
+        "title": "On the generating functions of totally positive sequences I",
+        "type": "article-journal",
+        "volume": "2"
+      },
+      "label": "ASW52"
+    },
+    "AkianGaubertSharify2017": {
+      "bibtex": "@article{AkianGaubertSharify2017,\n  author = {Akian, Marianne and Gaubert, Stéphane and Sharify, Meisam},\n  title = {Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots},\n  year = {2017},\n  journal = {Linear Algebra and its Applications},\n  volume = {528},\n  pages = {394--435},\n  doi = {10.1016/j.laa.2016.11.004},\n  url = {https://doi.org/10.1016/j.laa.2016.11.004}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.laa.2016.11.004",
+        "URL": "https://doi.org/10.1016/j.laa.2016.11.004",
+        "author": [
+          {
+            "family": "Akian",
+            "given": "Marianne"
+          },
+          {
+            "family": "Gaubert",
+            "given": "Stéphane"
+          },
+          {
+            "family": "Sharify",
+            "given": "Meisam"
+          }
+        ],
+        "container-title": "Linear Algebra and its Applications",
+        "issued": {
+          "date-parts": [
+            [
+              2017
+            ]
+          ]
+        },
+        "page": "394-435",
+        "title": "Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots",
+        "type": "article-journal",
+        "volume": "528"
+      },
+      "label": "AGS17"
+    },
+    "Alexandersson2026Eulerian": {
+      "bibtex": "@article{Alexandersson2026Eulerian,\n  author = {Per Alexandersson},\n  title = {Real-rooted {E}ulerian polynomials from permutations, words, and\n    paths},\n  year = {2026},\n  eprint = {2609.07325v1},\n  url = {https://arxiv.org/abs/2609.07325v1},\n  journal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/2609.07325v1",
+        "author": [
+          {
+            "family": "Alexandersson",
+            "given": "Per"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2026
+            ]
+          ]
+        },
+        "title": "Real-rooted Eulerian polynomials from permutations, words, and paths",
+        "type": "article-journal"
+      },
+      "label": "Ale26"
+    },
+    "AlexanderssonJal2024x": {
+      "bibtex": "@article{AlexanderssonJal2024x,\nAuthor = {Per Alexandersson and Aryaman Jal},\nTitle = {Rook matroids and log-concavity of ${P}$-{E}ulerian polynomials},\nYear = {2024},\nEprint = {2410.00127},\n  url = {https://arxiv.org/abs/2410.00127},\njournal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/2410.00127",
+        "author": [
+          {
+            "family": "Alexandersson",
+            "given": "Per"
+          },
+          {
+            "family": "Jal",
+            "given": "Aryaman"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2024
+            ]
+          ]
+        },
+        "title": "Rook matroids and log-concavity of ${P}$-Eulerian polynomials",
+        "type": "article-journal"
+      },
+      "label": "AJ24"
+    },
+    "AndrewsEggeGawronskiLittlejohn2013": {
+      "bibtex": "@article{AndrewsEggeGawronskiLittlejohn2013,\n  author = {Andrews, George E. and Egge, Eric S. and Gawronski, Wolfgang and Littlejohn, Lance L.},\n  title = {The {J}acobi–{S}tirling numbers},\n  year = {2013},\n  journal = {Journal of Combinatorial Theory, Series A},\n  volume = {120},\n  number = {1},\n  pages = {288--303},\n  doi = {10.1016/j.jcta.2012.08.006},\n  url = {https://doi.org/10.1016/j.jcta.2012.08.006}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.jcta.2012.08.006",
+        "URL": "https://doi.org/10.1016/j.jcta.2012.08.006",
+        "author": [
+          {
+            "family": "Andrews",
+            "given": "George E."
+          },
+          {
+            "family": "Egge",
+            "given": "Eric S."
+          },
+          {
+            "family": "Gawronski",
+            "given": "Wolfgang"
+          },
+          {
+            "family": "Littlejohn",
+            "given": "Lance L."
+          }
+        ],
+        "container-title": "Journal of Combinatorial Theory, Series A",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              2013
+            ]
+          ]
+        },
+        "page": "288-303",
+        "title": "The Jacobi–Stirling numbers",
+        "type": "article-journal",
+        "volume": "120"
+      },
+      "label": "AEGL13"
+    },
+    "AndrewsGawronskiLittlejohn2011": {
+      "bibtex": "@article{AndrewsGawronskiLittlejohn2011,\n  author = {Andrews, G.E. and Gawronski, W. and Littlejohn, L.L.},\n  title = {The {L}egendre–{S}tirling numbers},\n  year = {2011},\n  journal = {Discrete Mathematics},\n  volume = {311},\n  number = {14},\n  pages = {1255--1272},\n  doi = {10.1016/j.disc.2011.02.028},\n  url = {https://doi.org/10.1016/j.disc.2011.02.028}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.disc.2011.02.028",
+        "URL": "https://doi.org/10.1016/j.disc.2011.02.028",
+        "author": [
+          {
+            "family": "Andrews",
+            "given": "G. E."
+          },
+          {
+            "family": "Gawronski",
+            "given": "W."
+          },
+          {
+            "family": "Littlejohn",
+            "given": "L. L."
+          }
+        ],
+        "container-title": "Discrete Mathematics",
+        "issue": "14",
+        "issued": {
+          "date-parts": [
+            [
+              2011
+            ]
+          ]
+        },
+        "page": "1255-1272",
+        "title": "The Legendre–Stirling numbers",
+        "type": "article-journal",
+        "volume": "311"
+      },
+      "label": "AGL11"
+    },
+    "Asano1970": {
+      "bibtex": "@article{Asano1970,\n  author = {Asano, Taro},\n  title = {Lee-{Y}ang {T}heorem and the {G}riffiths {I}nequality for the {A}nisotropic {H}eisenberg {F}erromagnet},\n  year = {1970},\n  journal = {Physical Review Letters},\n  volume = {24},\n  number = {25},\n  pages = {1409--1411},\n  doi = {10.1103/PhysRevLett.24.1409},\n  url = {https://doi.org/10.1103/PhysRevLett.24.1409}\n}\n",
+      "csl": {
+        "DOI": "10.1103/PhysRevLett.24.1409",
+        "URL": "https://doi.org/10.1103/PhysRevLett.24.1409",
+        "author": [
+          {
+            "family": "Asano",
+            "given": "Taro"
+          }
+        ],
+        "container-title": "Physical Review Letters",
+        "issue": "25",
+        "issued": {
+          "date-parts": [
+            [
+              1970
+            ]
+          ]
+        },
+        "page": "1409-1411",
+        "title": "Lee-Yang Theorem and the Griffiths Inequality for the Anisotropic Heisenberg Ferromagnet",
+        "type": "article-journal",
+        "volume": "24"
+      },
+      "label": "Asa70"
+    },
+    "Athanasiadis2025EulerianTransformation": {
+      "bibtex": "@article{Athanasiadis2025EulerianTransformation,\n  author = {Athanasiadis, Christos A.},\n  title = {On the real-rootedness of the {E}ulerian transformation},\n  journal = {Journal of the London Mathematical Society},\n  volume = {111},\n  number = {2},\n  year = {2025},\n  doi = {10.1112/jlms.70083},\n  url2 = {https://doi.org/10.1112/jlms.70083},\n  eprint = {2302.00754},\n  archivePrefix = {arXiv}\n}\n",
+      "csl": {
+        "DOI": "10.1112/jlms.70083",
+        "URL": "https://arxiv.org/abs/2302.00754",
+        "author": [
+          {
+            "family": "Athanasiadis",
+            "given": "Christos A."
+          }
+        ],
+        "container-title": "Journal of the London Mathematical Society",
+        "issue": "2",
+        "issued": {
+          "date-parts": [
+            [
+              2025
+            ]
+          ]
+        },
+        "title": "On the real-rootedness of the Eulerian transformation",
+        "type": "article-journal",
+        "volume": "111"
+      },
+      "label": "Ath25"
+    },
+    "AthanasiadisWagner2024x": {
+      "bibtex": "@article{AthanasiadisWagner2024x,\n  author = {Christos A. Athanasiadis and David G. Wagner},\n  title = {Veronese sections and interlacing matrices of polynomials and formal power series},\n  journal = {arXiv e-prints},\n  year = {2024},\n  eprint = {2404.12989},\n  doi = {10.48550/arXiv.2404.12989}\n}\n",
+      "csl": {
+        "DOI": "10.48550/arXiv.2404.12989",
+        "URL": "https://arxiv.org/abs/2404.12989",
+        "author": [
+          {
+            "family": "Athanasiadis",
+            "given": "Christos A."
+          },
+          {
+            "family": "Wagner",
+            "given": "David G."
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2024
+            ]
+          ]
+        },
+        "title": "Veronese sections and interlacing matrices of polynomials and formal power series",
+        "type": "article-journal"
+      },
+      "label": "AW24"
+    },
+    "BarcucciLungoPinzani1996": {
+      "bibtex": "@article{BarcucciLungoPinzani1996,\n  author = {Barcucci, Elena and Del Lungo, Alberto and Pinzani, Renzo},\n  title = {“{D}eco” polyominoes, permutations and random generation},\n  year = {1996},\n  journal = {Theoretical Computer Science},\n  volume = {159},\n  number = {1},\n  pages = {29--42},\n  doi = {10.1016/0304-3975(95)00199-9},\n  url = {https://doi.org/10.1016/0304-3975(95)00199-9}\n}\n",
+      "csl": {
+        "DOI": "10.1016/0304-3975(95)00199-9",
+        "URL": "https://doi.org/10.1016/0304-3975(95)00199-9",
+        "author": [
+          {
+            "family": "Barcucci",
+            "given": "Elena"
+          },
+          {
+            "family": "Del Lungo",
+            "given": "Alberto"
+          },
+          {
+            "family": "Pinzani",
+            "given": "Renzo"
+          }
+        ],
+        "container-title": "Theoretical Computer Science",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              1996
+            ]
+          ]
+        },
+        "page": "29-42",
+        "title": "“Deco” polyominoes, permutations and random generation",
+        "type": "article-journal",
+        "volume": "159"
+      },
+      "label": "BDLP96"
+    },
+    "Bender1973": {
+      "bibtex": "@article{Bender1973,\n  title = {Central and local limit theorems applied to asymptotic enumeration},\n  volume = {15},\n  ISSN = {0097-3165},\n  url = {http://dx.doi.org/10.1016/0097-3165(73)90038-1},\n  DOI = {10.1016/0097-3165(73)90038-1},\n  number = {1},\n  journal = {Journal of Combinatorial Theory,  Series A},\n  publisher = {Elsevier BV},\n  author = {Bender,  Edward A},\n  year = {1973},\n  month = jul,\n  pages = {91–-111}\n}\n",
+      "csl": {
+        "DOI": "10.1016/0097-3165(73)90038-1",
+        "URL": "http://dx.doi.org/10.1016/0097-3165(73)90038-1",
+        "author": [
+          {
+            "family": "Bender",
+            "given": "Edward A"
+          }
+        ],
+        "container-title": "Journal of Combinatorial Theory, Series A",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              1973,
+              7
+            ]
+          ]
+        },
+        "page": "91--111",
+        "title": "Central and local limit theorems applied to asymptotic enumeration",
+        "type": "article-journal",
+        "volume": "15"
+      },
+      "label": "Ben73"
+    },
+    "BinderVecchi2026x": {
+      "bibtex": "@article{BinderVecchi2026x,\n  author = {Kyle Binder and Lorenzo Vecchi},\n  title = {Augmented singular cohomology, uniform matroids, and\n    real-rootedness},\n  year = {2026},\n  eprint = {2609.15946v1},\n  url = {https://arxiv.org/abs/2609.15946v1},\n  journal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/2609.15946v1",
+        "author": [
+          {
+            "family": "Binder",
+            "given": "Kyle"
+          },
+          {
+            "family": "Vecchi",
+            "given": "Lorenzo"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2026
+            ]
+          ]
+        },
+        "title": "Augmented singular cohomology, uniform matroids, and real-rootedness",
+        "type": "article-journal"
+      },
+      "label": "BV26"
+    },
+    "BogvadHagg2017": {
+      "bibtex": "@article{BogvadHagg2017,\n  author = {Bøgvad, Rikard and Hägg, Christian},\n  title = {A refinement for rational functions of {P}ólya's method to construct {V}oronoi diagrams},\n  year = {2017},\n  journal = {Journal of Mathematical Analysis and Applications},\n  volume = {452},\n  number = {1},\n  pages = {312--334},\n  doi = {10.1016/j.jmaa.2017.02.071},\n  url = {https://doi.org/10.1016/j.jmaa.2017.02.071}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.jmaa.2017.02.071",
+        "URL": "https://doi.org/10.1016/j.jmaa.2017.02.071",
+        "author": [
+          {
+            "family": "Bøgvad",
+            "given": "Rikard"
+          },
+          {
+            "family": "Hägg",
+            "given": "Christian"
+          }
+        ],
+        "container-title": "Journal of Mathematical Analysis and Applications",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              2017
+            ]
+          ]
+        },
+        "page": "312-334",
+        "title": "A refinement for rational functions of Pólya’s method to construct Voronoi diagrams",
+        "type": "article-journal",
+        "volume": "452"
+      },
+      "label": "BH17"
+    },
+    "Bona2009": {
+      "bibtex": "@article{Bona2009,\n  doi = {10.1137/070702254},\n  url = {https://doi.org/10.1137/070702254},\n  year = {2009},\n  month = jan,\n  publisher = {Society for Industrial \\& Applied Mathematics ({SIAM})},\n  volume = {23},\n  number = {1},\n  pages = {401--406},\n  author = {Mikl{\\'{o}}s B{\\'{o}}na},\n  title = {Real Zeros and Normal Distribution for Statistics on {S}tirling Permutations Defined by {G}essel and {S}tanley},\n  journal = {{SIAM} Journal on Discrete Mathematics}\n}\n",
+      "csl": {
+        "DOI": "10.1137/070702254",
+        "URL": "https://doi.org/10.1137/070702254",
+        "author": [
+          {
+            "family": "Bóna",
+            "given": "Miklós"
+          }
+        ],
+        "container-title": "SIAM Journal on Discrete Mathematics",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              2009,
+              1
+            ]
+          ]
+        },
+        "page": "401-406",
+        "title": "Real zeros and normal distribution for statistics on Stirling permutations defined by Gessel and Stanley",
+        "type": "article-journal",
+        "volume": "23"
+      },
+      "label": "Bo09"
+    },
+    "BonaHezo2016": {
+      "bibtex": "@article{BonaHezo2016,\n  doi = {10.1016/j.ejc.2015.07.021},\n  url = {https://doi.org/10.1016/j.ejc.2015.07.021},\n  year = {2016},\n  month = jan,\n  publisher = {Elsevier {BV}},\n  volume = {51},\n  pages = {500--510},\n  author = {Mikl{\\'{o}}s B{\\'{o}}na and Istv{\\'{a}}n Mez{\\H{o}}},\n  title = {Real zeros and partitions without singleton blocks},\n  journal = {European Journal of Combinatorics}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.ejc.2015.07.021",
+        "URL": "https://doi.org/10.1016/j.ejc.2015.07.021",
+        "author": [
+          {
+            "family": "Bóna",
+            "given": "Miklós"
+          },
+          {
+            "family": "Mező",
+            "given": "István"
+          }
+        ],
+        "container-title": "European Journal of Combinatorics",
+        "issued": {
+          "date-parts": [
+            [
+              2016,
+              1
+            ]
+          ]
+        },
+        "page": "500-510",
+        "title": "Real zeros and partitions without singleton blocks",
+        "type": "article-journal",
+        "volume": "51"
+      },
+      "label": "BM16"
+    },
+    "BorceaBranden2008MixedDeterminants": {
+      "bibtex": "@article{BorceaBranden2008MixedDeterminants,\n  doi = {10.1215/00127094-2008-018},\n  url2 = {https://doi.org/10.1215/00127094-2008-018},\n  year = {2008},\n  month = jun,\n  publisher = {Duke University Press},\n  volume = {143},\n  number = {2},\n  pages = {205--223},\n  author = {Julius Borcea and Petter Br{\\\"{a}}nd{\\'{e}}n},\n  title = {Applications of stable polynomials to mixed determinants: {J}ohnson's conjectures, unimodality, and symmetrized {F}ischer products},\n  journal = {Duke Mathematical Journal}\n}\n",
+      "csl": {
+        "DOI": "10.1215/00127094-2008-018",
+        "author": [
+          {
+            "family": "Borcea",
+            "given": "Julius"
+          },
+          {
+            "family": "Brändén",
+            "given": "Petter"
+          }
+        ],
+        "container-title": "Duke Mathematical Journal",
+        "issue": "2",
+        "issued": {
+          "date-parts": [
+            [
+              2008,
+              6
+            ]
+          ]
+        },
+        "page": "205-223",
+        "title": "Applications of stable polynomials to mixed determinants: Johnson’s conjectures, unimodality, and symmetrized Fischer products",
+        "type": "article-journal",
+        "volume": "143"
+      },
+      "label": "BB08"
+    },
+    "BorceaBranden2009I": {
+      "bibtex": "@article{BorceaBranden2009I,\n  doi = {10.1007/s00222-009-0189-3},\n  url2 = {https://doi.org/10.1007/s00222-009-0189-3},\n  year = {2009},\n  month = mar,\n  publisher = {Springer Science and Business Media {LLC}},\n  volume = {177},\n  number = {3},\n  pages = {541--569},\n  author = {Julius Borcea and Petter Br{\\\"{a}}nd{\\'{e}}n},\n  title = {The {L}ee--{Y}ang and {P}{\\'{o}}lya--{S}chur programs. {I}. {L}inear operators preserving stability},\n  journal = {Inventiones mathematicae}\n}\n",
+      "csl": {
+        "DOI": "10.1007/s00222-009-0189-3",
+        "author": [
+          {
+            "family": "Borcea",
+            "given": "Julius"
+          },
+          {
+            "family": "Brändén",
+            "given": "Petter"
+          }
+        ],
+        "container-title": "Inventiones mathematicae",
+        "issue": "3",
+        "issued": {
+          "date-parts": [
+            [
+              2009,
+              3
+            ]
+          ]
+        },
+        "page": "541-569",
+        "title": "The Lee–Yang and Pólya–Schur programs. I. Linear operators preserving stability",
+        "type": "article-journal",
+        "volume": "177"
+      },
+      "label": "BB09"
+    },
+    "BorceaBranden2009II": {
+      "bibtex": "@article{BorceaBranden2009II,\n\tdoi = {10.1002/cpa.20295},\n\turl2 = {https://doi.org/10.1002%2Fcpa.20295},\n\tyear = 2009,\n\tmonth = {dec},\n\tpublisher = {Wiley},\n\tvolume = {62},\n\tnumber = {12},\n\tpages = {1595--1631},\n\tauthor = {Julius Borcea and Petter Br{\\\"{a}}nd{\\'{e}}n},\n\ttitle = {The {L}ee--{Y}ang and {P}{\\'{o}}lya--{S}chur programs. {II}. Theory of stable polynomials and applications},\n\tjournal = {Communications on Pure and Applied Mathematics}\n}\n",
+      "csl": {
+        "DOI": "10.1002/cpa.20295",
+        "author": [
+          {
+            "family": "Borcea",
+            "given": "Julius"
+          },
+          {
+            "family": "Brändén",
+            "given": "Petter"
+          }
+        ],
+        "container-title": "Communications on Pure and Applied Mathematics",
+        "issue": "12",
+        "issued": {
+          "date-parts": [
+            [
+              2009,
+              12
+            ]
+          ]
+        },
+        "page": "1595-1631",
+        "title": "The Lee–Yang and Pólya–Schur programs. II. Theory of stable polynomials and applications",
+        "type": "article-journal",
+        "volume": "62"
+      },
+      "label": "BB09"
+    },
+    "BorceaBranden2010": {
+      "bibtex": "@article{BorceaBranden2010,\n  doi = {10.1112/plms/pdp049},\n  url = {https://doi.org/10.1112/plms/pdp049},\n  year = {2010},\n  month = jul,\n  publisher = {Wiley},\n  volume = {101},\n  number = {1},\n  pages = {73--104},\n  author = {J. Borcea and P. Br{\\\"{a}}nd{\\'{e}}n},\n  title = {Multivariate {P}{\\'{o}}lya--{S}chur classification problems in the {W}eyl algebra},\n  journal = {Proceedings of the London Mathematical Society}\n}\n",
+      "csl": {
+        "DOI": "10.1112/plms/pdp049",
+        "URL": "https://doi.org/10.1112/plms/pdp049",
+        "author": [
+          {
+            "family": "Borcea",
+            "given": "J."
+          },
+          {
+            "family": "Brändén",
+            "given": "P."
+          }
+        ],
+        "container-title": "Proceedings of the London Mathematical Society",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              2010,
+              7
+            ]
+          ]
+        },
+        "page": "73-104",
+        "title": "Multivariate Pólya–Schur classification problems in the Weyl algebra",
+        "type": "article-journal",
+        "volume": "101"
+      },
+      "label": "BB10"
+    },
+    "BorceaBrandenLiggett2008": {
+      "bibtex": "@article{BorceaBrandenLiggett2008,\n  title = {Negative dependence and the geometry of polynomials},\n  volume = {22},\n  ISSN = {1088-6834},\n  url = {http://dx.doi.org/10.1090/S0894-0347-08-00618-8},\n  DOI = {10.1090/s0894-0347-08-00618-8},\n  number = {2},\n  journal = {Journal of the American Mathematical Society},\n  publisher = {American Mathematical Society (AMS)},\n  author = {Borcea,  Julius and Br{\\\"{a}}nd{\\'{e}}n,  Petter and Liggett,  Thomas},\n  year = {2008},\n  month = sep,\n  pages = {521–567}\n}\n",
+      "csl": {
+        "DOI": "10.1090/s0894-0347-08-00618-8",
+        "URL": "http://dx.doi.org/10.1090/S0894-0347-08-00618-8",
+        "author": [
+          {
+            "family": "Borcea",
+            "given": "Julius"
+          },
+          {
+            "family": "Brändén",
+            "given": "Petter"
+          },
+          {
+            "family": "Liggett",
+            "given": "Thomas"
+          }
+        ],
+        "container-title": "Journal of the American Mathematical Society",
+        "issue": "2",
+        "issued": {
+          "date-parts": [
+            [
+              2008,
+              9
+            ]
+          ]
+        },
+        "page": "521-567",
+        "title": "Negative dependence and the geometry of polynomials",
+        "type": "article-journal",
+        "volume": "22"
+      },
+      "label": "BBL08"
+    },
+    "BorosMoll1999": {
+      "bibtex": "@article{BorosMoll1999,\n  author = {Boros, George and Moll, Victor H.},\n  title = {An integral hidden in {G}radshteyn and {R}yzhik},\n  year = {1999},\n  journal = {Journal of Computational and Applied Mathematics},\n  volume = {106},\n  number = {2},\n  pages = {361--368},\n  doi = {10.1016/S0377-0427(99)00081-3},\n  url = {https://doi.org/10.1016/S0377-0427(99)00081-3}\n}\n",
+      "csl": {
+        "DOI": "10.1016/S0377-0427(99)00081-3",
+        "URL": "https://doi.org/10.1016/S0377-0427(99)00081-3",
+        "author": [
+          {
+            "family": "Boros",
+            "given": "George"
+          },
+          {
+            "family": "Moll",
+            "given": "Victor H."
+          }
+        ],
+        "container-title": "Journal of Computational and Applied Mathematics",
+        "issue": "2",
+        "issued": {
+          "date-parts": [
+            [
+              1999
+            ]
+          ]
+        },
+        "page": "361-368",
+        "title": "An integral hidden in Gradshteyn and Ryzhik",
+        "type": "article-journal",
+        "volume": "106"
+      },
+      "label": "BM99"
+    },
+    "Branden2004operators": {
+      "bibtex": "@article{Branden2004operators,\n  title = {On Operators on Polynomials Preserving Real-Rootedness and the {N}eggers-{S}tanley Conjecture},\n  volume = {20},\n  ISSN = {0925-9899},\n  url2 = {http://dx.doi.org/10.1023/B:JACO.0000047295.93525.df},\n  DOI = {10.1023/b:jaco.0000047295.93525.df},\n  number = {2},\n  journal = {Journal of Algebraic Combinatorics},\n  publisher = {Springer Science and Business Media LLC},\n  author = {Petter Br{\\\"{a}}nd{\\'{e}}n},\n  year = {2004},\n  month = sep,\n  pages = {119–130}\n}\n",
+      "csl": {
+        "DOI": "10.1023/b:jaco.0000047295.93525.df",
+        "author": [
+          {
+            "family": "Brändén",
+            "given": "Petter"
+          }
+        ],
+        "container-title": "Journal of Algebraic Combinatorics",
+        "issue": "2",
+        "issued": {
+          "date-parts": [
+            [
+              2004,
+              9
+            ]
+          ]
+        },
+        "page": "119-130",
+        "title": "On operators on polynomials preserving real-rootedness and the Neggers-Stanley conjecture",
+        "type": "article-journal",
+        "volume": "20"
+      },
+      "label": "Br04"
+    },
+    "Branden2006": {
+      "bibtex": "@article{Branden2006,\n  title = {On linear transformations preserving the {P}{\\'{o}}lya frequency property},\n  volume = {358},\n  ISSN = {1088-6850},\n  url = {http://dx.doi.org/10.1090/S0002-9947-06-03856-6},\n  DOI = {10.1090/s0002-9947-06-03856-6},\n  number = {8},\n  journal = {Transactions of the American Mathematical Society},\n  publisher = {American Mathematical Society (AMS)},\n  author = {Br{\\\"a}nd{\\'e}n, Petter},\n  year = {2006},\n  month = feb,\n  pages = {3697--3716}\n}\n",
+      "csl": {
+        "DOI": "10.1090/s0002-9947-06-03856-6",
+        "URL": "http://dx.doi.org/10.1090/S0002-9947-06-03856-6",
+        "author": [
+          {
+            "family": "Brändén",
+            "given": "Petter"
+          }
+        ],
+        "container-title": "Transactions of the American Mathematical Society",
+        "issue": "8",
+        "issued": {
+          "date-parts": [
+            [
+              2006,
+              2
+            ]
+          ]
+        },
+        "page": "3697-3716",
+        "title": "On linear transformations preserving the Pólya frequency property",
+        "type": "article-journal",
+        "volume": "358"
+      },
+      "label": "Br06"
+    },
+    "Branden2007": {
+      "bibtex": "@article{Branden2007,\n  title={Polynomials with the half-plane property and matroid theory},\n  author={Br{\\\"{a}}nd{\\'{e}}n, Petter},\n  journal={Advances in Mathematics},\n  volume={216},\n  number={1},\n  pages={302--320},\n  year={2007},\n  doi={10.1016/j.aim.2007.05.011},\n  publisher={Elsevier}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.aim.2007.05.011",
+        "author": [
+          {
+            "family": "Brändén",
+            "given": "Petter"
+          }
+        ],
+        "container-title": "Advances in Mathematics",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              2007
+            ]
+          ]
+        },
+        "page": "302-320",
+        "title": "Polynomials with the half-plane property and matroid theory",
+        "type": "article-journal",
+        "volume": "216"
+      },
+      "label": "Br07"
+    },
+    "Branden2011b": {
+      "bibtex": "@article{Branden2011b,\n  author = {Brändén, Petter},\n  title = {Iterated sequences and the geometry of zeros},\n  year = {2011},\n  journal = {Journal für die reine und angewandte Mathematik (Crelles Journal)},\n  volume = {2011},\n  number = {658},\n  doi = {10.1515/crelle.2011.063},\n  url = {https://doi.org/10.1515/crelle.2011.063}\n}\n",
+      "csl": {
+        "DOI": "10.1515/crelle.2011.063",
+        "URL": "https://doi.org/10.1515/crelle.2011.063",
+        "author": [
+          {
+            "family": "Brändén",
+            "given": "Petter"
+          }
+        ],
+        "container-title": "Journal für die reine und angewandte Mathematik (Crelles Journal)",
+        "issue": "658",
+        "issued": {
+          "date-parts": [
+            [
+              2011
+            ]
+          ]
+        },
+        "title": "Iterated sequences and the geometry of zeros",
+        "type": "article-journal",
+        "volume": "2011"
+      },
+      "label": "Br11"
+    },
+    "Branden2015": {
+      "bibtex": "@incollection{Branden2015,\n  doi = {10.1201/b18255-10},\n  year = {2015},\n  month = mar,\n  publisher = {Chapman and Hall/{CRC}},\n  pages = {437--483},\n  author = {Petter Br{\\\"{a}}nd{\\'{e}}n},\n  title = {Unimodality, Log-Concavity, Real-Rootedness and Beyond},\n  booktitle = {Handbook of Enumerative Combinatorics}\n}\n",
+      "csl": {
+        "DOI": "10.1201/b18255-10",
+        "author": [
+          {
+            "family": "Brändén",
+            "given": "Petter"
+          }
+        ],
+        "container-title": "Handbook of enumerative combinatorics",
+        "issued": {
+          "date-parts": [
+            [
+              2015,
+              3
+            ]
+          ]
+        },
+        "page": "437-483",
+        "title": "Unimodality, log-concavity, real-rootedness and beyond",
+        "type": "chapter"
+      },
+      "label": "Br15"
+    },
+    "BrandenFerroniJochemko2024x": {
+      "bibtex": "@article{BrandenFerroniJochemko2024x,\n  author = {Petter Br{\\\"{a}}nd{\\'{e}}n and Luis Ferroni and Katharina Jochemko},\n  title = {Preservation of inequalities under {H}adamard products},\n  year = {2024},\n  eprint = {2408.12386v2},\n  url = {https://arxiv.org/abs/2408.12386v2},\n  journal = {Transactions of the American Mathematical Society},\n  note = {To appear}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/2408.12386v2",
+        "author": [
+          {
+            "family": "Brändén",
+            "given": "Petter"
+          },
+          {
+            "family": "Ferroni",
+            "given": "Luis"
+          },
+          {
+            "family": "Jochemko",
+            "given": "Katharina"
+          }
+        ],
+        "container-title": "Transactions of the American Mathematical Society",
+        "issued": {
+          "date-parts": [
+            [
+              2024
+            ]
+          ]
+        },
+        "title": "Preservation of inequalities under Hadamard products",
+        "type": "article-journal"
+      },
+      "label": "BFJ24"
+    },
+    "BrandenHaglundVisontaiWagner2011": {
+      "bibtex": "@incollection{BrandenHaglundVisontaiWagner2011,\n  doi = {10.1007/978-3-0348-0142-3_5},\n  url2 = {https://doi.org/10.1007/978-3-0348-0142-3_5},\n  year = {2011},\n  publisher = {Springer Basel},\n  pages = {63--78},\n  author = {Petter Br{\\\"{a}}nd{\\'{e}}n and James Haglund and Mirk{\\'{o}} Visontai and David G. Wagner},\n  title = {Proof of the Monotone Column Permanent Conjecture},\n  booktitle = {Notions of Positivity and the Geometry of Polynomials}\n}\n",
+      "csl": {
+        "DOI": "10.1007/978-3-0348-0142-3_5",
+        "author": [
+          {
+            "family": "Brändén",
+            "given": "Petter"
+          },
+          {
+            "family": "Haglund",
+            "given": "James"
+          },
+          {
+            "family": "Visontai",
+            "given": "Mirkó"
+          },
+          {
+            "family": "Wagner",
+            "given": "David G."
+          }
+        ],
+        "container-title": "Notions of positivity and the geometry of polynomials",
+        "issued": {
+          "date-parts": [
+            [
+              2011
+            ]
+          ]
+        },
+        "page": "63-78",
+        "title": "Proof of the monotone column permanent conjecture",
+        "type": "chapter"
+      },
+      "label": "BHVW11"
+    },
+    "BrandenHuh2020": {
+      "bibtex": "@article{BrandenHuh2020,\n  title = {Lorentzian polynomials},\n  volume = {192},\n  ISSN = {0003-486X},\n  url = {http://dx.doi.org/10.4007/annals.2020.192.3.4},\n  DOI = {10.4007/annals.2020.192.3.4},\n  number = {3},\n  journal = {Annals of Mathematics},\n  publisher = {Annals of Mathematics},\n  author = {Br{\\\"{a}}nd{\\'{e}}n,  Petter and Huh,  June},\n  year = {2020},\n  month = nov\n}\n",
+      "csl": {
+        "DOI": "10.4007/annals.2020.192.3.4",
+        "URL": "http://dx.doi.org/10.4007/annals.2020.192.3.4",
+        "author": [
+          {
+            "family": "Brändén",
+            "given": "Petter"
+          },
+          {
+            "family": "Huh",
+            "given": "June"
+          }
+        ],
+        "container-title": "Annals of Mathematics",
+        "issue": "3",
+        "issued": {
+          "date-parts": [
+            [
+              2020,
+              11
+            ]
+          ]
+        },
+        "title": "Lorentzian polynomials",
+        "type": "article-journal",
+        "volume": "192"
+      },
+      "label": "BH20"
+    },
+    "BrandenJochemko2022": {
+      "bibtex": "@article{BrandenJochemko2022,\n  author = {Petter Br{\\\"a}nd{\\'e}n and Katharina Jochemko},\n  title = {The {E}ulerian transformation},\n  journal = {Transactions of the American Mathematical Society},\n  volume = {375},\n  number = {3},\n  pages = {1917--1931},\n  year = {2022},\n  doi = {10.1090/tran/8539},\n  eprint = {2103.00890},\n  url = {https://arxiv.org/abs/2103.00890}\n}\n",
+      "csl": {
+        "DOI": "10.1090/tran/8539",
+        "URL": "https://arxiv.org/abs/2103.00890",
+        "author": [
+          {
+            "family": "Brändén",
+            "given": "Petter"
+          },
+          {
+            "family": "Jochemko",
+            "given": "Katharina"
+          }
+        ],
+        "container-title": "Transactions of the American Mathematical Society",
+        "issue": "3",
+        "issued": {
+          "date-parts": [
+            [
+              2022
+            ]
+          ]
+        },
+        "page": "1917-1931",
+        "title": "The Eulerian transformation",
+        "type": "article-journal",
+        "volume": "375"
+      },
+      "label": "BJ22"
+    },
+    "BrandenLeite2024x": {
+      "bibtex": "@article{BrandenLeite2024x,\n  author = {Petter Brändén and Leonardo Saud Maia Leite},\n  title = {Totally nonnegative matrices, chain enumeration and zeros of polynomials},\n  journal = {arXiv e-prints},\n  Eprint = {2412.06595},\n  url = {https://arxiv.org/abs/2412.06595},\n  year = {2024}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/2412.06595",
+        "author": [
+          {
+            "family": "Brändén",
+            "given": "Petter"
+          },
+          {
+            "family": "Leite",
+            "given": "Leonardo Saud Maia"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2024
+            ]
+          ]
+        },
+        "title": "Totally nonnegative matrices, chain enumeration and zeros of polynomials",
+        "type": "article-journal"
+      },
+      "label": "BL24"
+    },
+    "BrandenSolus2019": {
+      "bibtex": "@article{BrandenSolus2019,\n  title = {Symmetric Decompositions and Real-Rootedness},\n  volume = {2021},\n  ISSN = {1687-0247},\n  url = {http://dx.doi.org/10.1093/imrn/rnz059},\n  DOI = {10.1093/imrn/rnz059},\n  number = {10},\n  journal = {International Mathematics Research Notices},\n  publisher = {Oxford University Press (OUP)},\n  author = {Br\\\"{a}ndén, Petter and Solus, Liam},\n  year = {2019},\n  month = apr,\n  pages = {7764--7798}\n}\n",
+      "csl": {
+        "DOI": "10.1093/imrn/rnz059",
+        "URL": "http://dx.doi.org/10.1093/imrn/rnz059",
+        "author": [
+          {
+            "family": "Brändén",
+            "given": "Petter"
+          },
+          {
+            "family": "Solus",
+            "given": "Liam"
+          }
+        ],
+        "container-title": "International Mathematics Research Notices",
+        "issue": "10",
+        "issued": {
+          "date-parts": [
+            [
+              2019,
+              4
+            ]
+          ]
+        },
+        "page": "7764-7798",
+        "title": "Symmetric decompositions and real-rootedness",
+        "type": "article-journal",
+        "volume": "2021"
+      },
+      "label": "BS19"
+    },
+    "BrandenVecchi2025x": {
+      "bibtex": "@article{BrandenVecchi2025x,\n  author = {Petter Brändén and Lorenzo Vecchi},\n  title = {Chow polynomials of totally nonnegative matrices and posets},\n  year = {2025},\n  eprint = {2509.17852v3},\n  url = {https://arxiv.org/abs/2509.17852v3},\n  journal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/2509.17852v3",
+        "author": [
+          {
+            "family": "Brändén",
+            "given": "Petter"
+          },
+          {
+            "family": "Vecchi",
+            "given": "Lorenzo"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2025
+            ]
+          ]
+        },
+        "title": "Chow polynomials of totally nonnegative matrices and posets",
+        "type": "article-journal"
+      },
+      "label": "BV25"
+    },
+    "BraunJal2026x": {
+      "bibtex": "@article{BraunJal2026x,\n  author = {Benjamin Braun and Aryaman Jal},\n  title = {Order polytopes of generalized snake posets are {$h^*$}-real-rooted},\n  year = {2026},\n  eprint = {2607.00922},\n  url = {https://arxiv.org/abs/2607.00922},\n  journal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/2607.00922",
+        "author": [
+          {
+            "family": "Braun",
+            "given": "Benjamin"
+          },
+          {
+            "family": "Jal",
+            "given": "Aryaman"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2026
+            ]
+          ]
+        },
+        "title": "Order polytopes of generalized snake posets are $h^*$-real-rooted",
+        "type": "article-journal"
+      },
+      "label": "BJ26"
+    },
+    "Brenti1989": {
+      "bibtex": "@article{Brenti1989,\n    AUTHOR = {Brenti, Francesco},\n     TITLE = {Unimodal, log-concave and {P}{\\'{o}}lya frequency sequences in\n              combinatorics},\n   JOURNAL = {Mem. Amer. Math. Soc.},\n  FJOURNAL = {Memoirs of the American Mathematical Society},\n    VOLUME = {81},\n      YEAR = {1989},\n    NUMBER = {413},\n     PAGES = {viii+106},\n      ISSN = {0065-9266,1947-6221},\n   MRCLASS = {05A15 (05A10 05A20)},\n  MRNUMBER = {963833},\nMRREVIEWER = {Ira\\ Gessel},\n       DOI = {10.1090/memo/0413},\n       URL = {https://doi.org/10.1090/memo/0413},\n}\n",
+      "csl": {
+        "DOI": "10.1090/memo/0413",
+        "URL": "https://doi.org/10.1090/memo/0413",
+        "author": [
+          {
+            "family": "Brenti",
+            "given": "Francesco"
+          }
+        ],
+        "container-title": "Mem. Amer. Math. Soc.",
+        "issue": "413",
+        "issued": {
+          "date-parts": [
+            [
+              1989
+            ]
+          ]
+        },
+        "page": "viii+106",
+        "title": "Unimodal, log-concave and Pólya frequency sequences in combinatorics",
+        "type": "article-journal",
+        "volume": "81"
+      },
+      "label": "Bre89"
+    },
+    "Brenti1994": {
+      "bibtex": "@article{Brenti1994,\n  author = {Brenti, Francesco},\n  title = {q-{E}ulerian {P}olynomials {A}rising from {C}oxeter {G}roups},\n  year = {1994},\n  journal = {European Journal of Combinatorics},\n  volume = {15},\n  number = {5},\n  pages = {417--441},\n  doi = {10.1006/eujc.1994.1046},\n  url = {https://doi.org/10.1006/eujc.1994.1046}\n}\n",
+      "csl": {
+        "DOI": "10.1006/eujc.1994.1046",
+        "URL": "https://doi.org/10.1006/eujc.1994.1046",
+        "author": [
+          {
+            "family": "Brenti",
+            "given": "Francesco"
+          }
+        ],
+        "container-title": "European Journal of Combinatorics",
+        "issue": "5",
+        "issued": {
+          "date-parts": [
+            [
+              1994
+            ]
+          ]
+        },
+        "page": "417-441",
+        "title": "Q-Eulerian Polynomials Arising from Coxeter Groups",
+        "type": "article-journal",
+        "volume": "15"
+      },
+      "label": "Bre94"
+    },
+    "ChoeOxleySokalWagner2004": {
+      "bibtex": "@article{ChoeOxleySokalWagner2004,\n  doi = {10.1016/s0196-8858(03)00078-2},\n  url2 = {https://doi.org/10.1016/s0196-8858(03)00078-2},\n  year = {2004},\n  month = jan,\n  publisher = {Elsevier {BV}},\n  volume = {32},\n  number = {1-2},\n  pages = {88--187},\n  author = {Young-Bin Choe and James G. Oxley and Alan D. Sokal and David G. Wagner},\n  title = {Homogeneous multivariate polynomials with the half-plane property},\n  journal = {Advances in Applied Mathematics}\n}\n",
+      "csl": {
+        "DOI": "10.1016/s0196-8858(03)00078-2",
+        "author": [
+          {
+            "family": "Choe",
+            "given": "Young-Bin"
+          },
+          {
+            "family": "Oxley",
+            "given": "James G."
+          },
+          {
+            "family": "Sokal",
+            "given": "Alan D."
+          },
+          {
+            "family": "Wagner",
+            "given": "David G."
+          }
+        ],
+        "container-title": "Advances in Applied Mathematics",
+        "issue": "1-2",
+        "issued": {
+          "date-parts": [
+            [
+              2004,
+              1
+            ]
+          ]
+        },
+        "page": "88-187",
+        "title": "Homogeneous multivariate polynomials with the half-plane property",
+        "type": "article-journal",
+        "volume": "32"
+      },
+      "label": "COSW04"
+    },
+    "ChowShiu2011": {
+      "bibtex": "@article{ChowShiu2011,\n  doi = {10.1007/s00026-011-0113-6},\n  url2 = {https://doi.org/10.1007/s00026-011-0113-6},\n  year = {2011},\n  month = oct,\n  publisher = {Springer Science and Business Media {LLC}},\n  volume = {15},\n  number = {4},\n  pages = {625--635},\n  author = {Chak-On Chow and Wai Chee Shiu},\n  title = {Counting Simsun Permutations by Descents},\n  journal = {Annals of Combinatorics}\n}\n",
+      "csl": {
+        "DOI": "10.1007/s00026-011-0113-6",
+        "author": [
+          {
+            "family": "Chow",
+            "given": "Chak-On"
+          },
+          {
+            "family": "Shiu",
+            "given": "Wai Chee"
+          }
+        ],
+        "container-title": "Annals of Combinatorics",
+        "issue": "4",
+        "issued": {
+          "date-parts": [
+            [
+              2011,
+              10
+            ]
+          ]
+        },
+        "page": "625-635",
+        "title": "Counting simsun permutations by descents",
+        "type": "article-journal",
+        "volume": "15"
+      },
+      "label": "CS11"
+    },
+    "ChudnovskySeymour2007": {
+      "bibtex": "@article{ChudnovskySeymour2007,\n  doi = {10.1016/j.jctb.2006.06.001},\n  url2 = {https://doi.org/10.1016/j.jctb.2006.06.001},\n  year = {2007},\n  month = may,\n  publisher = {Elsevier {BV}},\n  volume = {97},\n  number = {3},\n  pages = {350--357},\n  author = {Maria Chudnovsky and Paul Seymour},\n  title = {The roots of the independence polynomial of a clawfree graph},\n  journal = {Journal of Combinatorial Theory,  Series B}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.jctb.2006.06.001",
+        "author": [
+          {
+            "family": "Chudnovsky",
+            "given": "Maria"
+          },
+          {
+            "family": "Seymour",
+            "given": "Paul"
+          }
+        ],
+        "container-title": "Journal of Combinatorial Theory, Series B",
+        "issue": "3",
+        "issued": {
+          "date-parts": [
+            [
+              2007,
+              5
+            ]
+          ]
+        },
+        "page": "350-357",
+        "title": "The roots of the independence polynomial of a clawfree graph",
+        "type": "article-journal",
+        "volume": "97"
+      },
+      "label": "CS07"
+    },
+    "Cohn1922": {
+      "bibtex": "@article{Cohn1922,\n  author = {Cohn, A.},\n  title = {{\\\"U}ber die {A}nzahl der {W}urzeln einer algebraischen {G}leichung in einem {K}reise},\n  year = {1922},\n  journal = {Mathematische Zeitschrift},\n  volume = {14},\n  number = {1},\n  pages = {110--148},\n  doi = {10.1007/BF01215894},\n  url = {https://doi.org/10.1007/BF01215894}\n}\n",
+      "csl": {
+        "DOI": "10.1007/BF01215894",
+        "URL": "https://doi.org/10.1007/BF01215894",
+        "author": [
+          {
+            "family": "Cohn",
+            "given": "A."
+          }
+        ],
+        "container-title": "Mathematische Zeitschrift",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              1922
+            ]
+          ]
+        },
+        "page": "110-148",
+        "title": "Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise",
+        "type": "article-journal",
+        "volume": "14"
+      },
+      "label": "Coh22"
+    },
+    "Courant1920": {
+      "bibtex": "@article{Courant1920,\n  author = {Courant, R.},\n  title = {{\\\"U}ber die {E}igenwerte bei den {D}ifferentialgleichungen der mathematischen {P}hysik},\n  year = {1920},\n  journal = {Mathematische Zeitschrift},\n  volume = {7},\n  number = {1-4},\n  pages = {1--57},\n  doi = {10.1007/BF01199396},\n  url = {https://doi.org/10.1007/BF01199396}\n}\n",
+      "csl": {
+        "DOI": "10.1007/BF01199396",
+        "URL": "https://doi.org/10.1007/BF01199396",
+        "author": [
+          {
+            "family": "Courant",
+            "given": "R."
+          }
+        ],
+        "container-title": "Mathematische Zeitschrift",
+        "issue": "1-4",
+        "issued": {
+          "date-parts": [
+            [
+              1920
+            ]
+          ]
+        },
+        "page": "1-57",
+        "title": "Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik",
+        "type": "article-journal",
+        "volume": "7"
+      },
+      "label": "Cou20"
+    },
+    "Cryer1976": {
+      "bibtex": "@article{Cryer1976,\n  author = {Cryer, Colin W.},\n  title = {Some properties of totally positive matrices},\n  year = {1976},\n  journal = {Linear Algebra and its Applications},\n  volume = {15},\n  number = {1},\n  pages = {1--25},\n  doi = {10.1016/0024-3795(76)90076-8},\n  url = {https://doi.org/10.1016/0024-3795(76)90076-8}\n}\n",
+      "csl": {
+        "DOI": "10.1016/0024-3795(76)90076-8",
+        "URL": "https://doi.org/10.1016/0024-3795(76)90076-8",
+        "author": [
+          {
+            "family": "Cryer",
+            "given": "Colin W."
+          }
+        ],
+        "container-title": "Linear Algebra and its Applications",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              1976
+            ]
+          ]
+        },
+        "page": "1-25",
+        "title": "Some properties of totally positive matrices",
+        "type": "article-journal",
+        "volume": "15"
+      },
+      "label": "Cry76"
+    },
+    "Dedieu1992": {
+      "bibtex": "@article{Dedieu1992,\n  author = {Dedieu, Jean-Pierre},\n  title = {Obreschkoff's theorem revisited: What convex sets are contained in the set of hyperbolic polynomials?},\n  journal = {Journal of Pure and Applied Algebra},\n  volume = {81},\n  number = {3},\n  pages = {269--278},\n  year = {1992}\n}\n",
+      "csl": {
+        "author": [
+          {
+            "family": "Dedieu",
+            "given": "Jean-Pierre"
+          }
+        ],
+        "container-title": "Journal of Pure and Applied Algebra",
+        "issue": "3",
+        "issued": {
+          "date-parts": [
+            [
+              1992
+            ]
+          ]
+        },
+        "page": "269-278",
+        "title": "Obreschkoff’s theorem revisited: What convex sets are contained in the set of hyperbolic polynomials?",
+        "type": "article-journal",
+        "volume": "81"
+      },
+      "label": "Ded92"
+    },
+    "DiaconisHicks2017": {
+      "bibtex": "@article{DiaconisHicks2017,\nauthor = {Persi Diaconis and Angela Hicks},\ntitle = {Probabilizing parking functions},\njournal = {Advances in Applied Mathematics},\nvolume = {89},\npages = {125--155},\nyear = {2017},\ndoi = {10.1016/j.aam.2017.05.004},\nurl = {https://doi.org/10.1016/j.aam.2017.05.004},\neprint = {1611.09821}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.aam.2017.05.004",
+        "URL": "https://arxiv.org/abs/1611.09821",
+        "author": [
+          {
+            "family": "Diaconis",
+            "given": "Persi"
+          },
+          {
+            "family": "Hicks",
+            "given": "Angela"
+          }
+        ],
+        "container-title": "Advances in Applied Mathematics",
+        "issued": {
+          "date-parts": [
+            [
+              2017
+            ]
+          ]
+        },
+        "page": "125-155",
+        "title": "Probabilizing parking functions",
+        "type": "article-journal",
+        "volume": "89"
+      },
+      "label": "DH17"
+    },
+    "DominiciJohnstonJordaan2013x": {
+      "bibtex": "@article{DominiciJohnstonJordaan2013x,\n  author = {D. Dominici and S. J. Johnston and K. Jordaan},\n  title = {Real zeros of 2{F}1 hypergeometric polynomials},\n  year = {2013},\n  eprint = {1301.4771v1},\n  url = {https://arxiv.org/abs/1301.4771v1},\n  journal = {arXiv e-prints},\n  doi = {10.1016/j.cam.2012.12.024}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.cam.2012.12.024",
+        "URL": "https://arxiv.org/abs/1301.4771v1",
+        "author": [
+          {
+            "family": "Dominici",
+            "given": "D."
+          },
+          {
+            "family": "Johnston",
+            "given": "S. J."
+          },
+          {
+            "family": "Jordaan",
+            "given": "K."
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2013
+            ]
+          ]
+        },
+        "title": "Real zeros of 2F1 hypergeometric polynomials",
+        "type": "article-journal"
+      },
+      "label": "DJJ13"
+    },
+    "Edrei1953": {
+      "bibtex": "@article{Edrei1953,\n  author = {Edrei, Albert},\n  title = {Proof of a {C}onjecture of {S}choenberg on the {G}enerating {F}unction of a {T}otally {P}ositive {S}equence},\n  year = {1953},\n  journal = {Canadian Journal of Mathematics},\n  volume = {5},\n  pages = {86--94},\n  doi = {10.4153/CJM-1953-010-3},\n  url = {https://doi.org/10.4153/CJM-1953-010-3}\n}\n",
+      "csl": {
+        "DOI": "10.4153/CJM-1953-010-3",
+        "URL": "https://doi.org/10.4153/CJM-1953-010-3",
+        "author": [
+          {
+            "family": "Edrei",
+            "given": "Albert"
+          }
+        ],
+        "container-title": "Canadian Journal of Mathematics",
+        "issued": {
+          "date-parts": [
+            [
+              1953
+            ]
+          ]
+        },
+        "page": "86-94",
+        "title": "Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence",
+        "type": "article-journal",
+        "volume": "5"
+      },
+      "label": "Edr53"
+    },
+    "Egorychev1981": {
+      "bibtex": "@article{Egorychev1981,\n  author = {Egorychev, G.P},\n  title = {The solution of van der {W}aerden's problem for permanents},\n  year = {1981},\n  journal = {Advances in Mathematics},\n  volume = {42},\n  number = {3},\n  pages = {299--305},\n  doi = {10.1016/0001-8708(81)90044-X},\n  url = {https://doi.org/10.1016/0001-8708(81)90044-X}\n}\n",
+      "csl": {
+        "DOI": "10.1016/0001-8708(81)90044-X",
+        "URL": "https://doi.org/10.1016/0001-8708(81)90044-X",
+        "author": [
+          {
+            "family": "Egorychev",
+            "given": "G. P"
+          }
+        ],
+        "container-title": "Advances in Mathematics",
+        "issue": "3",
+        "issued": {
+          "date-parts": [
+            [
+              1981
+            ]
+          ]
+        },
+        "page": "299-305",
+        "title": "The solution of van der Waerden’s problem for permanents",
+        "type": "article-journal",
+        "volume": "42"
+      },
+      "label": "Ego81"
+    },
+    "ElizaldeRiveraZhuang2026": {
+      "bibtex": "@article{ElizaldeRiveraZhuang2026,\n  author = {Elizalde, Sergi and Rivera, Johnny and Zhuang, Yan},\n  title = {Counting {P}attern-{A}voiding {P}ermutations by {B}ig {D}escents},\n  year = {2026},\n  journal = {Annals of Combinatorics},\n  publisher = {Springer Science and Business Media LLC},\n  doi = {10.1007/s00026-026-00805-1},\n  url = {http://dx.doi.org/10.1007/s00026-026-00805-1},\n  issn = {0219-3094}\n}\n",
+      "csl": {
+        "DOI": "10.1007/s00026-026-00805-1",
+        "URL": "http://dx.doi.org/10.1007/s00026-026-00805-1",
+        "author": [
+          {
+            "family": "Elizalde",
+            "given": "Sergi"
+          },
+          {
+            "family": "Rivera",
+            "given": "Johnny"
+          },
+          {
+            "family": "Zhuang",
+            "given": "Yan"
+          }
+        ],
+        "container-title": "Annals of Combinatorics",
+        "issued": {
+          "date-parts": [
+            [
+              2026
+            ]
+          ]
+        },
+        "title": "Counting Pattern-Avoiding Permutations by Big Descents",
+        "type": "article-journal"
+      },
+      "label": "ERZ26"
+    },
+    "Engstrom2007": {
+      "bibtex": "@article{Engstrom2007,\nauthor = {Engström, Alexander},\nyear = {2007},\nmonth = {01},\npages = {1--10},\nurl = {https://emis.de/ft/14440},\ntitle = {Inequalities on well-distributed point sets on circles},\nvolume = {8},\njournal = {Journal of Inequalities in Pure and Applied Mathematics}\n}\n",
+      "csl": {
+        "URL": "https://emis.de/ft/14440",
+        "author": [
+          {
+            "family": "Engström",
+            "given": "Alexander"
+          }
+        ],
+        "container-title": "Journal of Inequalities in Pure and Applied Mathematics",
+        "issued": {
+          "date-parts": [
+            [
+              2007,
+              1
+            ]
+          ]
+        },
+        "page": "1-10",
+        "title": "Inequalities on well-distributed point sets on circles",
+        "type": "article-journal",
+        "volume": "8"
+      },
+      "label": "Eng07"
+    },
+    "Falikman1981": {
+      "bibtex": "@article{Falikman1981,\n  author = {Falikman, D. I.},\n  title = {Proof of the van der {W}aerden conjecture regarding the permanent of a doubly stochastic matrix},\n  year = {1981},\n  journal = {Mathematical Notes of the Academy of Sciences of the USSR},\n  volume = {29},\n  number = {6},\n  pages = {475--479},\n  doi = {10.1007/BF01163285},\n  url = {https://doi.org/10.1007/BF01163285}\n}\n",
+      "csl": {
+        "DOI": "10.1007/BF01163285",
+        "URL": "https://doi.org/10.1007/BF01163285",
+        "author": [
+          {
+            "family": "Falikman",
+            "given": "D. I."
+          }
+        ],
+        "container-title": "Mathematical Notes of the Academy of Sciences of the USSR",
+        "issue": "6",
+        "issued": {
+          "date-parts": [
+            [
+              1981
+            ]
+          ]
+        },
+        "page": "475-479",
+        "title": "Proof of the van der Waerden conjecture regarding the permanent of a doubly stochastic matrix",
+        "type": "article-journal",
+        "volume": "29"
+      },
+      "label": "Fal81"
+    },
+    "FallatJohnson2011": {
+      "bibtex": "@book{FallatJohnson2011,\n  author    = {Fallat, Shaun M. and Johnson, Charles R.},\n  title     = {Totally Nonnegative Matrices},\n  publisher = {Princeton University Press},\n  series    = {Princeton Series in Applied Mathematics},\n  year      = {2011},\n  doi       = {10.1515/9781400839018}\n}\n",
+      "csl": {
+        "DOI": "10.1515/9781400839018",
+        "author": [
+          {
+            "family": "Fallat",
+            "given": "Shaun M."
+          },
+          {
+            "family": "Johnson",
+            "given": "Charles R."
+          }
+        ],
+        "issued": {
+          "date-parts": [
+            [
+              2011
+            ]
+          ]
+        },
+        "title": "Totally nonnegative matrices",
+        "type": "book"
+      },
+      "label": "FJ11"
+    },
+    "Favard1935": {
+      "bibtex": "@article{Favard1935,\n  author = {Jean Favard},\n  title = {Sur les polyn{\\^o}mes de Tchebicheff},\n  journal = {C. R. Acad. Sci. Paris},\n  volume = {200},\n  year = {1935},\n  pages = {2052--2053}\n}\n",
+      "csl": {
+        "author": [
+          {
+            "family": "Favard",
+            "given": "Jean"
+          }
+        ],
+        "container-title": "C. R. Acad. Sci. Paris",
+        "issued": {
+          "date-parts": [
+            [
+              1935
+            ]
+          ]
+        },
+        "page": "2052-2053",
+        "title": "Sur les polynômes de tchebicheff",
+        "type": "article-journal",
+        "volume": "200"
+      },
+      "label": "Fav35"
+    },
+    "Fischer1905": {
+      "bibtex": "@article{Fischer1905,\n  author = {Fischer, Ernst},\n  title = {Über quadratische {F}ormen mit reellen {K}oeffizienten},\n  year = {1905},\n  journal = {Monatshefte für Mathematik und Physik},\n  volume = {16},\n  number = {1},\n  pages = {234--249},\n  doi = {10.1007/BF01693781},\n  url = {https://doi.org/10.1007/BF01693781}\n}\n",
+      "csl": {
+        "DOI": "10.1007/BF01693781",
+        "URL": "https://doi.org/10.1007/BF01693781",
+        "author": [
+          {
+            "family": "Fischer",
+            "given": "Ernst"
+          }
+        ],
+        "container-title": "Monatshefte für Mathematik und Physik",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              1905
+            ]
+          ]
+        },
+        "page": "234-249",
+        "title": "Über quadratische Formen mit reellen Koeffizienten",
+        "type": "article-journal",
+        "volume": "16"
+      },
+      "label": "Fis05"
+    },
+    "Fisk2005x": {
+      "bibtex": "@article{Fisk2005x,\n  author = {Steve Fisk},\n  title = {A very short proof of {C}auchy's interlace theorem for eigenvalues of {H}ermitian matrices},\n  year = {2005},\n  eprint = {math/0502408},\n  url = {https://arxiv.org/abs/math/0502408},\n  journal = {arXiv e-prints},\n  journalref = {Amer. Math. Monthly, Vol 112, No. 2, Feb. 2005, page 118}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/math/0502408",
+        "author": [
+          {
+            "family": "Fisk",
+            "given": "Steve"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2005
+            ]
+          ]
+        },
+        "title": "A very short proof of Cauchy’s interlace theorem for eigenvalues of Hermitian matrices",
+        "type": "article-journal"
+      },
+      "label": "Fis05"
+    },
+    "Fisk2006x": {
+      "bibtex": "@article{Fisk2006x,\n  Author = {Steve Fisk},\n  Title = {Polynomials, roots, and interlacing},\n  Year = {2006},\n  Eprint = {arXiv:math/0612833},\n  journal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/math/0612833",
+        "author": [
+          {
+            "family": "Fisk",
+            "given": "Steve"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2006
+            ]
+          ]
+        },
+        "title": "Polynomials, roots, and interlacing",
+        "type": "article-journal"
+      },
+      "label": "Fis06"
+    },
+    "FuLinZeng2018": {
+      "bibtex": "@article{FuLinZeng2018,\n  author = {Fu, Shishuo and Lin, Zhicong and Zeng, Jiang},\n  title = {On two unimodal descent polynomials},\n  year = {2018},\n  journal = {Discrete Mathematics},\n  volume = {341},\n  number = {9},\n  pages = {2616--2626},\n  publisher = {Elsevier BV},\n  doi = {10.1016/j.disc.2018.06.010},\n  url = {https://doi.org/10.1016/j.disc.2018.06.010},\n  issn = {0012-365X}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.disc.2018.06.010",
+        "URL": "https://doi.org/10.1016/j.disc.2018.06.010",
+        "author": [
+          {
+            "family": "Fu",
+            "given": "Shishuo"
+          },
+          {
+            "family": "Lin",
+            "given": "Zhicong"
+          },
+          {
+            "family": "Zeng",
+            "given": "Jiang"
+          }
+        ],
+        "container-title": "Discrete Mathematics",
+        "issue": "9",
+        "issued": {
+          "date-parts": [
+            [
+              2018
+            ]
+          ]
+        },
+        "page": "2616-2626",
+        "title": "On two unimodal descent polynomials",
+        "type": "article-journal",
+        "volume": "341"
+      },
+      "label": "FLZ18"
+    },
+    "Garding1959": {
+      "bibtex": "@article{Garding1959,\n  author = {Gårding, Lars},\n  title = {An {I}nequality for {H}yperbolic {P}olynomials},\n  year = {1959},\n  journal = {Indiana University Mathematics Journal},\n  volume = {8},\n  number = {6},\n  pages = {957--965},\n  doi = {10.1512/iumj.1959.8.58061},\n  url = {https://doi.org/10.1512/iumj.1959.8.58061}\n}\n",
+      "csl": {
+        "DOI": "10.1512/iumj.1959.8.58061",
+        "URL": "https://doi.org/10.1512/iumj.1959.8.58061",
+        "author": [
+          {
+            "family": "Gårding",
+            "given": "Lars"
+          }
+        ],
+        "container-title": "Indiana University Mathematics Journal",
+        "issue": "6",
+        "issued": {
+          "date-parts": [
+            [
+              1959
+            ]
+          ]
+        },
+        "page": "957-965",
+        "title": "An Inequality for Hyperbolic Polynomials",
+        "type": "article-journal",
+        "volume": "8"
+      },
+      "label": "Ga59"
+    },
+    "GarloffWagner1996": {
+      "bibtex": "@article{GarloffWagner1996,\n  doi = {10.1006/jmaa.1996.0348},\n  url2 = {https://doi.org/10.1006/jmaa.1996.0348},\n  year = {1996},\n  publisher = {Elsevier {BV}},\n  volume = {202},\n  number = {3},\n  pages = {797--809},\n  author = {J{\\\"u}rgen Garloff and David G. Wagner},\n  title = {Hadamard Products of Stable Polynomials Are Stable},\n  journal = {Journal of Mathematical Analysis and Applications}\n}\n",
+      "csl": {
+        "DOI": "10.1006/jmaa.1996.0348",
+        "author": [
+          {
+            "family": "Garloff",
+            "given": "Jürgen"
+          },
+          {
+            "family": "Wagner",
+            "given": "David G."
+          }
+        ],
+        "container-title": "Journal of Mathematical Analysis and Applications",
+        "issue": "3",
+        "issued": {
+          "date-parts": [
+            [
+              1996
+            ]
+          ]
+        },
+        "page": "797-809",
+        "title": "Hadamard products of stable polynomials are stable",
+        "type": "article-journal",
+        "volume": "202"
+      },
+      "label": "GW96"
+    },
+    "GelineauZeng2010": {
+      "bibtex": "@article{GelineauZeng2010,\n  author = {Gelineau, Yoann and Zeng, Jiang},\n  title = {Combinatorial {I}nterpretations of the {J}acobi-{S}tirling {N}umbers},\n  year = {2010},\n  journal = {The Electronic Journal of Combinatorics},\n  volume = {17},\n  number = {1},\n  doi = {10.37236/342},\n  url = {https://doi.org/10.37236/342}\n}\n",
+      "csl": {
+        "DOI": "10.37236/342",
+        "URL": "https://doi.org/10.37236/342",
+        "author": [
+          {
+            "family": "Gelineau",
+            "given": "Yoann"
+          },
+          {
+            "family": "Zeng",
+            "given": "Jiang"
+          }
+        ],
+        "container-title": "The Electronic Journal of Combinatorics",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              2010
+            ]
+          ]
+        },
+        "title": "Combinatorial Interpretations of the Jacobi-Stirling Numbers",
+        "type": "article-journal",
+        "volume": "17"
+      },
+      "label": "GZ10"
+    },
+    "GesselLinZeng2012": {
+      "bibtex": "@article{GesselLinZeng2012,\n  author = {Gessel, Ira M. and Lin, Zhicong and Zeng, Jiang},\n  title = {Jacobi--{S}tirling polynomials and {$P$}-partitions},\n  year = {2012},\n  journal = {European Journal of Combinatorics},\n  volume = {33},\n  number = {8},\n  pages = {1987--2000},\n  doi = {10.1016/j.ejc.2012.06.008},\n  eprint = {1201.0622v2},\n  url = {https://doi.org/10.1016/j.ejc.2012.06.008}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.ejc.2012.06.008",
+        "URL": "https://doi.org/10.1016/j.ejc.2012.06.008",
+        "author": [
+          {
+            "family": "Gessel",
+            "given": "Ira M."
+          },
+          {
+            "family": "Lin",
+            "given": "Zhicong"
+          },
+          {
+            "family": "Zeng",
+            "given": "Jiang"
+          }
+        ],
+        "container-title": "European Journal of Combinatorics",
+        "issue": "8",
+        "issued": {
+          "date-parts": [
+            [
+              2012
+            ]
+          ]
+        },
+        "page": "1987-2000",
+        "title": "Jacobi–Stirling polynomials and $P$-partitions",
+        "type": "article-journal",
+        "volume": "33"
+      },
+      "label": "GLZ12"
+    },
+    "GesselStanley1978": {
+      "bibtex": "@article{GesselStanley1978,\n  doi = {10.1016/0097-3165(78)90042-0},\n  url2 = {https://doi.org/10.1016/0097-3165(78)90042-0},\n  year = {1978},\n  month = jan,\n  publisher = {Elsevier {BV}},\n  volume = {24},\n  number = {1},\n  pages = {24--33},\n  author = {Ira Gessel and Richard P Stanley},\n  title = {Stirling polynomials},\n  journal = {Journal of Combinatorial Theory,  Series A}\n}\n",
+      "csl": {
+        "DOI": "10.1016/0097-3165(78)90042-0",
+        "author": [
+          {
+            "family": "Gessel",
+            "given": "Ira"
+          },
+          {
+            "family": "Stanley",
+            "given": "Richard P"
+          }
+        ],
+        "container-title": "Journal of Combinatorial Theory, Series A",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              1978,
+              1
+            ]
+          ]
+        },
+        "page": "24-33",
+        "title": "Stirling polynomials",
+        "type": "article-journal",
+        "volume": "24"
+      },
+      "label": "GS78"
+    },
+    "GesselViennot1985": {
+      "bibtex": "@article{GesselViennot1985,\n  doi = {10.1016/0001-8708(85)90121-5},\n  url2 = {https://doi.org/10.1016/0001-8708(85)90121-5},\n  year = {1985},\n  month = dec,\n  publisher = {Elsevier {BV}},\n  volume = {58},\n  number = {3},\n  pages = {300--321},\n  author = {Ira Gessel and G{\\'{e}}rard Viennot},\n  title = {Binomial determinants,  paths, and hook length formulae},\n  journal = {Advances in Mathematics}\n}\n",
+      "csl": {
+        "DOI": "10.1016/0001-8708(85)90121-5",
+        "author": [
+          {
+            "family": "Gessel",
+            "given": "Ira"
+          },
+          {
+            "family": "Viennot",
+            "given": "Gérard"
+          }
+        ],
+        "container-title": "Advances in Mathematics",
+        "issue": "3",
+        "issued": {
+          "date-parts": [
+            [
+              1985,
+              12
+            ]
+          ]
+        },
+        "page": "300-321",
+        "title": "Binomial determinants, paths, and hook length formulae",
+        "type": "article-journal",
+        "volume": "58"
+      },
+      "label": "GV85"
+    },
+    "Godsil2017": {
+      "bibtex": "@book{Godsil2017,\n  doi = {10.1201/9781315137131},\n  url2 = {https://doi.org/10.1201/9781315137131},\n  year = {2017},\n  month = oct,\n  publisher = {Routledge},\n  author = {C.D. Godsil},\n  title = {Algebraic Combinatorics}\n}\n",
+      "csl": {
+        "DOI": "10.1201/9781315137131",
+        "author": [
+          {
+            "family": "Godsil",
+            "given": "C. D."
+          }
+        ],
+        "issued": {
+          "date-parts": [
+            [
+              2017,
+              10
+            ]
+          ]
+        },
+        "title": "Algebraic combinatorics",
+        "type": "book"
+      },
+      "label": "God17"
+    },
+    "Grace1902": {
+      "bibtex": "@article{Grace1902,\n  author  = {J. H. Grace},\n  title   = {The zeros of a polynomial},\n  journal = {Proceedings of the Cambridge Philosophical Society},\n  volume  = {11},\n  pages   = {352--357},\n  year    = {1902}\n}\n",
+      "csl": {
+        "author": [
+          {
+            "family": "Grace",
+            "given": "J. H."
+          }
+        ],
+        "container-title": "Proceedings of the Cambridge Philosophical Society",
+        "issued": {
+          "date-parts": [
+            [
+              1902
+            ]
+          ]
+        },
+        "page": "352-357",
+        "title": "The zeros of a polynomial",
+        "type": "article-journal",
+        "volume": "11"
+      },
+      "label": "Gra02"
+    },
+    "GribinskiMarcus2022": {
+      "bibtex": "@article{GribinskiMarcus2022,\n  author = {Gribinski, Aurelien and Marcus, Adam W.},\n  title = {A rectangular additive convolution for polynomials},\n  year = {2022},\n  journal = {Combinatorial Theory},\n  volume = {2},\n  number = {1},\n  doi = {10.5070/c62156888},\n  url = {https://doi.org/10.5070/c62156888}\n}\n",
+      "csl": {
+        "DOI": "10.5070/c62156888",
+        "URL": "https://doi.org/10.5070/c62156888",
+        "author": [
+          {
+            "family": "Gribinski",
+            "given": "Aurelien"
+          },
+          {
+            "family": "Marcus",
+            "given": "Adam W."
+          }
+        ],
+        "container-title": "Combinatorial Theory",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              2022
+            ]
+          ]
+        },
+        "title": "A rectangular additive convolution for polynomials",
+        "type": "article-journal",
+        "volume": "2"
+      },
+      "label": "GM22"
+    },
+    "Gurvits2008": {
+      "bibtex": "@article{Gurvits2008,\n  author = {Leonid Gurvits},\n  title = {Van der {W}aerden/{S}chrijver--{V}aliant like conjectures and stable\n    (aka hyperbolic) homogeneous polynomials: One theorem for all},\n  year = {2008},\n  journal = {The Electronic Journal of Combinatorics},\n  volume = {15},\n  number = {1},\n  pages = {Research Paper 66},\n  doi = {10.37236/790},\n  url = {https://doi.org/10.37236/790},\n  eprint = {0711.3496}\n}\n",
+      "csl": {
+        "DOI": "10.37236/790",
+        "URL": "https://arxiv.org/abs/0711.3496",
+        "author": [
+          {
+            "family": "Gurvits",
+            "given": "Leonid"
+          }
+        ],
+        "container-title": "The Electronic Journal of Combinatorics",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              2008
+            ]
+          ]
+        },
+        "page": "Research Paper 66",
+        "title": "Van der Waerden/Schrijver–Valiant like conjectures and stable (aka hyperbolic) homogeneous polynomials: One theorem for all",
+        "type": "article-journal",
+        "volume": "15"
+      },
+      "label": "Gur08"
+    },
+    "Gurvits2009": {
+      "bibtex": "@article{Gurvits2009,\n  author = {Gurvits, Leonid},\n  title = {A {S}hort {P}roof, {B}ased on {M}ixed {V}olumes, of {L}iggett's {T}heorem on the {C}onvolution of {U}ltra-{L}ogconcave {S}equences},\n  year = {2009},\n  journal = {The Electronic Journal of Combinatorics},\n  volume = {16},\n  number = {1},\n  doi = {10.37236/243},\n  url = {https://doi.org/10.37236/243}\n}\n",
+      "csl": {
+        "DOI": "10.37236/243",
+        "URL": "https://doi.org/10.37236/243",
+        "author": [
+          {
+            "family": "Gurvits",
+            "given": "Leonid"
+          }
+        ],
+        "container-title": "The Electronic Journal of Combinatorics",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              2009
+            ]
+          ]
+        },
+        "title": "A Short Proof, Based on Mixed Volumes, of Liggett’s Theorem on the Convolution of Ultra-Logconcave Sequences",
+        "type": "article-journal",
+        "volume": "16"
+      },
+      "label": "Gur09"
+    },
+    "GustafssonSolus2020": {
+      "bibtex": "@article{GustafssonSolus2020,\n  doi = {10.1016/j.aim.2020.107169},\n  url2 = {https://doi.org/10.1016/j.aim.2020.107169},\n  year = {2020},\n  month = aug,\n  publisher = {Elsevier {BV}},\n  volume = {369},\n  pages = {107169},\n  author = {Nils Gustafsson and Liam Solus},\n  title = {Derangements,  {E}hrhart theory,  and local h-polynomials},\n  journal = {Advances in Mathematics}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.aim.2020.107169",
+        "author": [
+          {
+            "family": "Gustafsson",
+            "given": "Nils"
+          },
+          {
+            "family": "Solus",
+            "given": "Liam"
+          }
+        ],
+        "container-title": "Advances in Mathematics",
+        "issued": {
+          "date-parts": [
+            [
+              2020,
+              8
+            ]
+          ]
+        },
+        "page": "107169",
+        "title": "Derangements, Ehrhart theory, and local h-polynomials",
+        "type": "article-journal",
+        "volume": "369"
+      },
+      "label": "GS20"
+    },
+    "HaglundOnoWagner1999": {
+      "bibtex": "@incollection{HaglundOnoWagner1999,\n  doi = {10.1007/978-1-4613-0305-3_13},\n  url = {https://doi.org/10.1007/978-1-4613-0305-3_13},\n  year = {1999},\n  publisher = {Springer {US}},\n  pages = {207--221},\n  author = {James Haglund and Ken Ono and David G. Wagner},\n  title = {Theorems and Conjectures Involving Rook Polynomials with Only Real Zeros},\n  booktitle = {Topics in Number Theory}\n}\n",
+      "csl": {
+        "DOI": "10.1007/978-1-4613-0305-3_13",
+        "URL": "https://doi.org/10.1007/978-1-4613-0305-3_13",
+        "author": [
+          {
+            "family": "Haglund",
+            "given": "James"
+          },
+          {
+            "family": "Ono",
+            "given": "Ken"
+          },
+          {
+            "family": "Wagner",
+            "given": "David G."
+          }
+        ],
+        "container-title": "Topics in number theory",
+        "issued": {
+          "date-parts": [
+            [
+              1999
+            ]
+          ]
+        },
+        "page": "207-221",
+        "title": "Theorems and conjectures involving rook polynomials with only real zeros",
+        "type": "chapter"
+      },
+      "label": "HOW99"
+    },
+    "HaglundZhang2019": {
+      "bibtex": "@article{HaglundZhang2019,\n  doi = {10.1016/j.aam.2019.05.001},\n  url = {https://doi.org/10.1016/j.aam.2019.05.001},\n  year = {2019},\n  month = aug,\n  publisher = {Elsevier {BV}},\n  volume = {109},\n  pages = {38--54},\n  author = {James Haglund and Philip B. Zhang},\n  title = {Real-rootedness of variations of {E}ulerian polynomials},\n  journal = {Advances in Applied Mathematics}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.aam.2019.05.001",
+        "URL": "https://doi.org/10.1016/j.aam.2019.05.001",
+        "author": [
+          {
+            "family": "Haglund",
+            "given": "James"
+          },
+          {
+            "family": "Zhang",
+            "given": "Philip B."
+          }
+        ],
+        "container-title": "Advances in Applied Mathematics",
+        "issued": {
+          "date-parts": [
+            [
+              2019,
+              8
+            ]
+          ]
+        },
+        "page": "38-54",
+        "title": "Real-rootedness of variations of Eulerian polynomials",
+        "type": "article-journal",
+        "volume": "109"
+      },
+      "label": "HZ19"
+    },
+    "Harper1967": {
+      "bibtex": "@article{Harper1967,\n  author = {Harper, L. H.},\n  title = {Stirling {B}ehavior is {A}symptotically {N}ormal},\n  year = {1967},\n  journal = {The Annals of Mathematical Statistics},\n  volume = {38},\n  number = {2},\n  pages = {410--414},\n  doi = {10.1214/aoms/1177698956},\n  url = {https://doi.org/10.1214/aoms/1177698956}\n}\n",
+      "csl": {
+        "DOI": "10.1214/aoms/1177698956",
+        "URL": "https://doi.org/10.1214/aoms/1177698956",
+        "author": [
+          {
+            "family": "Harper",
+            "given": "L. H."
+          }
+        ],
+        "container-title": "The Annals of Mathematical Statistics",
+        "issue": "2",
+        "issued": {
+          "date-parts": [
+            [
+              1967
+            ]
+          ]
+        },
+        "page": "410-414",
+        "title": "Stirling Behavior is Asymptotically Normal",
+        "type": "article-journal",
+        "volume": "38"
+      },
+      "label": "Har67"
+    },
+    "HeilmannLieb1972": {
+      "bibtex": "@article{HeilmannLieb1972,\n  doi = {10.1007/bf01877590},\n  url2 = {https://doi.org/10.1007/bf01877590},\n  year = {1972},\n  month = sep,\n  publisher = {Springer Science and Business Media {LLC}},\n  volume = {25},\n  number = {3},\n  pages = {190--232},\n  author = {Ole J. Heilmann and Elliott H. Lieb},\n  title = {Theory of monomer-dimer systems},\n  journal = {Communications in Mathematical Physics}\n}\n",
+      "csl": {
+        "DOI": "10.1007/bf01877590",
+        "author": [
+          {
+            "family": "Heilmann",
+            "given": "Ole J."
+          },
+          {
+            "family": "Lieb",
+            "given": "Elliott H."
+          }
+        ],
+        "container-title": "Communications in Mathematical Physics",
+        "issue": "3",
+        "issued": {
+          "date-parts": [
+            [
+              1972,
+              9
+            ]
+          ]
+        },
+        "page": "190-232",
+        "title": "Theory of monomer-dimer systems",
+        "type": "article-journal",
+        "volume": "25"
+      },
+      "label": "HL72"
+    },
+    "Holtz2003": {
+      "bibtex": "@article{Holtz2003,\n  author = {Holtz, Olga},\n  title = {Hermite--{B}iehler, {R}outh--{H}urwitz, and total positivity},\n  year = {2003},\n  journal = {Linear Algebra and its Applications},\n  volume = {372},\n  pages = {105--110},\n  publisher = {Elsevier BV},\n  doi = {10.1016/s0024-3795(03)00501-9},\n  url = {http://dx.doi.org/10.1016/S0024-3795(03)00501-9},\n  issn = {0024-3795}\n}\n",
+      "csl": {
+        "DOI": "10.1016/s0024-3795(03)00501-9",
+        "URL": "http://dx.doi.org/10.1016/S0024-3795(03)00501-9",
+        "author": [
+          {
+            "family": "Holtz",
+            "given": "Olga"
+          }
+        ],
+        "container-title": "Linear Algebra and its Applications",
+        "issued": {
+          "date-parts": [
+            [
+              2003
+            ]
+          ]
+        },
+        "page": "105-110",
+        "title": "Hermite–Biehler, Routh–Hurwitz, and total positivity",
+        "type": "article-journal",
+        "volume": "372"
+      },
+      "label": "Hol03"
+    },
+    "HosterStump2025x": {
+      "bibtex": "@article{HosterStump2025x,\n  author = {Elena Hoster and Christian Stump},\n  title = {Chow polynomials of simplicial posets with positive $h$-vector are\n    real-rooted},\n  year = {2025},\n  eprint = {2508.15538},\n  url = {https://arxiv.org/abs/2508.15538},\n  journal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/2508.15538",
+        "author": [
+          {
+            "family": "Hoster",
+            "given": "Elena"
+          },
+          {
+            "family": "Stump",
+            "given": "Christian"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2025
+            ]
+          ]
+        },
+        "title": "Chow polynomials of simplicial posets with positive $h$-vector are real-rooted",
+        "type": "article-journal"
+      },
+      "label": "HS25"
+    },
+    "Hutchinson1923": {
+      "bibtex": "@article{Hutchinson1923,\n  author = {Hutchinson, J. I.},\n  title = {On a remarkable class of entire functions},\n  year = {1923},\n  journal = {Transactions of the American Mathematical Society},\n  volume = {25},\n  number = {3},\n  pages = {325–332},\n  publisher = {American Mathematical Society (AMS)},\n  doi = {10.1090/s0002-9947-1923-1501248-1},\n  url = {http://dx.doi.org/10.1090/s0002-9947-1923-1501248-1},\n  issn = {0002-9947}\n}\n",
+      "csl": {
+        "DOI": "10.1090/s0002-9947-1923-1501248-1",
+        "URL": "http://dx.doi.org/10.1090/s0002-9947-1923-1501248-1",
+        "author": [
+          {
+            "family": "Hutchinson",
+            "given": "J. I."
+          }
+        ],
+        "container-title": "Transactions of the American Mathematical Society",
+        "issue": "3",
+        "issued": {
+          "date-parts": [
+            [
+              1923
+            ]
+          ]
+        },
+        "page": "325-332",
+        "title": "On a remarkable class of entire functions",
+        "type": "article-journal",
+        "volume": "25"
+      },
+      "label": "Hut23"
+    },
+    "Jensen1913": {
+      "bibtex": "@article{Jensen1913,\n  author = {Jensen, J. L. W. V.},\n  title = {Recherches sur la théorie des équations},\n  year = {1913},\n  journal = {Acta Mathematica},\n  volume = {36},\n  number = {0},\n  pages = {181--195},\n  doi = {10.1007/BF02422380},\n  url = {https://doi.org/10.1007/BF02422380}\n}\n",
+      "csl": {
+        "DOI": "10.1007/BF02422380",
+        "URL": "https://doi.org/10.1007/BF02422380",
+        "author": [
+          {
+            "family": "Jensen",
+            "given": "J. L. W. V."
+          }
+        ],
+        "container-title": "Acta Mathematica",
+        "issue": "0",
+        "issued": {
+          "date-parts": [
+            [
+              1913
+            ]
+          ]
+        },
+        "page": "181-195",
+        "title": "Recherches sur la théorie des équations",
+        "type": "article-journal",
+        "volume": "36"
+      },
+      "label": "Jen13"
+    },
+    "Karlin1968": {
+      "bibtex": "@book{Karlin1968,\n  author    = {Karlin, Samuel},\n  title     = {Total Positivity, {V}ol.~{I}},\n  publisher = {Stanford University Press},\n  year      = {1968}\n}\n",
+      "csl": {
+        "author": [
+          {
+            "family": "Karlin",
+            "given": "Samuel"
+          }
+        ],
+        "issued": {
+          "date-parts": [
+            [
+              1968
+            ]
+          ]
+        },
+        "title": "Total positivity, Vol. I",
+        "type": "book"
+      },
+      "label": "Kar68"
+    },
+    "KarpenkoVishnyakova2017": {
+      "bibtex": "@article{KarpenkoVishnyakova2017,\n  author = {Karpenko, Irina and Vishnyakova, Anna M.},\n  title = {On sufficient conditions for a polynomial to be sign-independently hyperbolic or to have real separated zeros},\n  year = {2017},\n  journal = {Mathematical Inequalities &amp; Applications},\n  number = {1},\n  pages = {237--245},\n  doi = {10.7153/mia-20-18},\n  url = {https://doi.org/10.7153/mia-20-18}\n}\n",
+      "csl": {
+        "DOI": "10.7153/mia-20-18",
+        "URL": "https://doi.org/10.7153/mia-20-18",
+        "author": [
+          {
+            "family": "Karpenko",
+            "given": "Irina"
+          },
+          {
+            "family": "Vishnyakova",
+            "given": "Anna M."
+          }
+        ],
+        "container-title": "Mathematical Inequalities &amp; Applications",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              2017
+            ]
+          ]
+        },
+        "page": "237-245",
+        "title": "On sufficient conditions for a polynomial to be sign-independently hyperbolic or to have real separated zeros",
+        "type": "article-journal"
+      },
+      "label": "KV17"
+    },
+    "Kostov2010": {
+      "bibtex": "@article{Kostov2010,\n  author = {Kostov, Vladimir P.},\n  title = {Interlacing properties and the {S}chur--{S}zeg{\\H{o}} composition},\n  year = {2010},\n  journal = {Functional Analysis and Other Mathematics},\n  volume = {3},\n  number = {1},\n  pages = {65--74},\n  doi = {10.1007/s11853-010-0039-2},\n  url = {https://doi.org/10.1007/s11853-010-0039-2}\n}\n",
+      "csl": {
+        "DOI": "10.1007/s11853-010-0039-2",
+        "URL": "https://doi.org/10.1007/s11853-010-0039-2",
+        "author": [
+          {
+            "family": "Kostov",
+            "given": "Vladimir P."
+          }
+        ],
+        "container-title": "Functional Analysis and Other Mathematics",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              2010
+            ]
+          ]
+        },
+        "page": "65-74",
+        "title": "Interlacing properties and the Schur–Szegő composition",
+        "type": "article-journal",
+        "volume": "3"
+      },
+      "label": "Kos10"
+    },
+    "KostovShapiro2006": {
+      "bibtex": "@article{KostovShapiro2006,\n  author = {Kostov, Vladimir and Shapiro, Boris},\n  title = {On the {S}chur--{S}zeg{\\\"o} composition of polynomials},\n  year = {2006},\n  journal = {Comptes Rendus. Mathématique},\n  volume = {343},\n  number = {2},\n  pages = {81--86},\n  doi = {10.1016/j.crma.2006.06.007},\n  url = {https://doi.org/10.1016/j.crma.2006.06.007}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.crma.2006.06.007",
+        "URL": "https://doi.org/10.1016/j.crma.2006.06.007",
+        "author": [
+          {
+            "family": "Kostov",
+            "given": "Vladimir"
+          },
+          {
+            "family": "Shapiro",
+            "given": "Boris"
+          }
+        ],
+        "container-title": "Comptes Rendus. Mathématique",
+        "issue": "2",
+        "issued": {
+          "date-parts": [
+            [
+              2006
+            ]
+          ]
+        },
+        "page": "81-86",
+        "title": "On the Schur–Szegö composition of polynomials",
+        "type": "article-journal",
+        "volume": "343"
+      },
+      "label": "KS06"
+    },
+    "Kozlov2022": {
+      "bibtex": "@article{Kozlov2022,\n  author = {Kozlov, Dmitry N.},\n  title = {Stirling complexes},\n  year = {2022},\n  journal = {Journal of Applied and Computational Topology},\n  volume = {7},\n  number = {1},\n  pages = {57--74},\n  doi = {10.1007/s41468-022-00096-4},\n  url = {https://doi.org/10.1007/s41468-022-00096-4}\n}\n",
+      "csl": {
+        "DOI": "10.1007/s41468-022-00096-4",
+        "URL": "https://doi.org/10.1007/s41468-022-00096-4",
+        "author": [
+          {
+            "family": "Kozlov",
+            "given": "Dmitry N."
+          }
+        ],
+        "container-title": "Journal of Applied and Computational Topology",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              2022
+            ]
+          ]
+        },
+        "page": "57-74",
+        "title": "Stirling complexes",
+        "type": "article-journal",
+        "volume": "7"
+      },
+      "label": "Koz22"
+    },
+    "Kurtz1992": {
+      "bibtex": "@article{Kurtz1992,\n  author = {Kurtz, David C.},\n  title = {A Sufficient Condition for All the Roots of a Polynomial To Be Real},\n  year = {1992},\n  journal = {The American Mathematical Monthly},\n  volume = {99},\n  number = {3},\n  pages = {259--263},\n  publisher = {Informa UK Limited},\n  doi = {10.1080/00029890.1992.11995845},\n  url = {http://dx.doi.org/10.1080/00029890.1992.11995845},\n  issn = {1930-0972}\n}\n",
+      "csl": {
+        "DOI": "10.1080/00029890.1992.11995845",
+        "URL": "http://dx.doi.org/10.1080/00029890.1992.11995845",
+        "author": [
+          {
+            "family": "Kurtz",
+            "given": "David C."
+          }
+        ],
+        "container-title": "The American Mathematical Monthly",
+        "issue": "3",
+        "issued": {
+          "date-parts": [
+            [
+              1992
+            ]
+          ]
+        },
+        "page": "259-263",
+        "title": "A sufficient condition for all the roots of a polynomial to be real",
+        "type": "article-journal",
+        "volume": "99"
+      },
+      "label": "Kur92"
+    },
+    "LeakeRyder2019": {
+      "bibtex": "@article{LeakeRyder2019,\n\tauthor = {Leake, Jonathan D. and Ryder, Nick R.},\n\ttitle = {Generalizations of the Matching Polynomial to the Multivariate Independence Polynomial},\n\tjournal = {Algebraic Combinatorics},\n\tpublisher = {MathOA foundation},\n\tvolume = {2},\n\tnumber = {5},\n\tyear = {2019},\n\tpages = {781-802},\n\tdoi = {10.5802/alco.63},\n\tlanguage = {en},\n\turl={alco.centre-mersenne.org/item/ALCO_2019__2_5_781_0/}\n}\n",
+      "csl": {
+        "DOI": "10.5802/alco.63",
+        "URL": "alco.centre-mersenne.org/item/ALCO_2019__2_5_781_0/",
+        "author": [
+          {
+            "family": "Leake",
+            "given": "Jonathan D."
+          },
+          {
+            "family": "Ryder",
+            "given": "Nick R."
+          }
+        ],
+        "container-title": "Algebraic Combinatorics",
+        "issue": "5",
+        "issued": {
+          "date-parts": [
+            [
+              2019
+            ]
+          ]
+        },
+        "page": "781-802",
+        "title": "Generalizations of the matching polynomial to the multivariate independence polynomial",
+        "type": "article-journal",
+        "volume": "2"
+      },
+      "label": "LR19"
+    },
+    "Leander2016x": {
+      "bibtex": "@article{Leander2016x,\n  author = {Madeleine Leander},\n  title = {Compatible polynomials and edgewise subdivisions},\n  year = {2016},\n  eprint = {1605.05287v1},\n  url = {https://arxiv.org/abs/1605.05287v1},\n  journal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/1605.05287v1",
+        "author": [
+          {
+            "family": "Leander",
+            "given": "Madeleine"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2016
+            ]
+          ]
+        },
+        "title": "Compatible polynomials and edgewise subdivisions",
+        "type": "article-journal"
+      },
+      "label": "Lea16"
+    },
+    "LeeYang1952": {
+      "bibtex": "@article{LeeYang1952,\n  author = {T. D. Lee and C. N. Yang},\n  title = {Statistical theory of equations of state and phase transitions.\n    {II}. Lattice gas and {I}sing model},\n  year = {1952},\n  journal = {Physical Review},\n  volume = {87},\n  number = {3},\n  pages = {410--419},\n  doi = {10.1103/PhysRev.87.410},\n  url = {https://doi.org/10.1103/PhysRev.87.410}\n}\n",
+      "csl": {
+        "DOI": "10.1103/PhysRev.87.410",
+        "URL": "https://doi.org/10.1103/PhysRev.87.410",
+        "author": [
+          {
+            "family": "Lee",
+            "given": "T. D."
+          },
+          {
+            "family": "Yang",
+            "given": "C. N."
+          }
+        ],
+        "container-title": "Physical Review",
+        "issue": "3",
+        "issued": {
+          "date-parts": [
+            [
+              1952
+            ]
+          ]
+        },
+        "page": "410-419",
+        "title": "Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model",
+        "type": "article-journal",
+        "volume": "87"
+      },
+      "label": "LY52"
+    },
+    "Levin1964": {
+      "bibtex": "@book{Levin1964,\n  author = {Levin, B. Ja.},\n  title = {Distribution of Zeros of Entire Functions},\n  series = {Translations of Mathematical Monographs},\n  volume = {5},\n  publisher = {American Mathematical Society},\n  address = {Providence, RI},\n  year = {1964},\n  doi = {10.1090/mmono/005},\n  url2 = {https://doi.org/10.1090/mmono/005}\n}\n",
+      "csl": {
+        "DOI": "10.1090/mmono/005",
+        "author": [
+          {
+            "family": "Levin",
+            "given": "B. Ja."
+          }
+        ],
+        "issued": {
+          "date-parts": [
+            [
+              1964
+            ]
+          ]
+        },
+        "title": "Distribution of zeros of entire functions",
+        "type": "book",
+        "volume": "5"
+      },
+      "label": "Lev64"
+    },
+    "LiebSokal1981": {
+      "bibtex": "@article{LiebSokal1981,\n  author = {Lieb, Elliott H. and Sokal, Alan D.},\n  title = {A general {L}ee-{Y}ang theorem for one-component and multicomponent ferromagnets},\n  year = {1981},\n  journal = {Communications in Mathematical Physics},\n  volume = {80},\n  number = {2},\n  pages = {153--179},\n  doi = {10.1007/bf01213009},\n  url = {https://doi.org/10.1007/bf01213009}\n}\n",
+      "csl": {
+        "DOI": "10.1007/bf01213009",
+        "URL": "https://doi.org/10.1007/bf01213009",
+        "author": [
+          {
+            "family": "Lieb",
+            "given": "Elliott H."
+          },
+          {
+            "family": "Sokal",
+            "given": "Alan D."
+          }
+        ],
+        "container-title": "Communications in Mathematical Physics",
+        "issue": "2",
+        "issued": {
+          "date-parts": [
+            [
+              1981
+            ]
+          ]
+        },
+        "page": "153-179",
+        "title": "A general Lee-Yang theorem for one-component and multicomponent ferromagnets",
+        "type": "article-journal",
+        "volume": "80"
+      },
+      "label": "LS81"
+    },
+    "LinMaMaZhou2021": {
+      "bibtex": "@article{LinMaMaZhou2021,\n  author = {Lin, Zhicong and Ma, Jun and Ma, Shi-Mei and Zhou, Yanghongbo},\n  title = {Weakly increasing trees on a multiset},\n  year = {2021},\n  journal = {Advances in Applied Mathematics},\n  volume = {129},\n  pages = {102206},\n  doi = {10.1016/j.aam.2021.102206},\n  url = {https://doi.org/10.1016/j.aam.2021.102206}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.aam.2021.102206",
+        "URL": "https://doi.org/10.1016/j.aam.2021.102206",
+        "author": [
+          {
+            "family": "Lin",
+            "given": "Zhicong"
+          },
+          {
+            "family": "Ma",
+            "given": "Jun"
+          },
+          {
+            "family": "Ma",
+            "given": "Shi-Mei"
+          },
+          {
+            "family": "Zhou",
+            "given": "Yanghongbo"
+          }
+        ],
+        "container-title": "Advances in Applied Mathematics",
+        "issued": {
+          "date-parts": [
+            [
+              2021
+            ]
+          ]
+        },
+        "page": "102206",
+        "title": "Weakly increasing trees on a multiset",
+        "type": "article-journal",
+        "volume": "129"
+      },
+      "label": "LMMZ21"
+    },
+    "Lindstrom1973": {
+      "bibtex": "@article{Lindstrom1973,\n  title = {On the Vector Representations of Induced Matroids},\n  volume = {5},\n  ISSN = {0024-6093},\n  url = {http://dx.doi.org/10.1112/blms/5.1.85},\n  DOI = {10.1112/blms/5.1.85},\n  number = {1},\n  journal = {Bulletin of the London Mathematical Society},\n  publisher = {Wiley},\n  author = {Lindstr\\\"{o}m,  Bernt},\n  year = {1973},\n  month = mar,\n  pages = {85–90}\n}\n",
+      "csl": {
+        "DOI": "10.1112/blms/5.1.85",
+        "URL": "http://dx.doi.org/10.1112/blms/5.1.85",
+        "author": [
+          {
+            "family": "Lindström",
+            "given": "Bernt"
+          }
+        ],
+        "container-title": "Bulletin of the London Mathematical Society",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              1973,
+              3
+            ]
+          ]
+        },
+        "page": "85-90",
+        "title": "On the vector representations of induced matroids",
+        "type": "article-journal",
+        "volume": "5"
+      },
+      "label": "Lin73"
+    },
+    "Liu2012CompatibleSequences": {
+      "bibtex": "@article{Liu2012CompatibleSequences,\n  author = {Liu, Lily L.},\n  title = {Polynomials with real zeros and compatible sequences},\n  year = {2012},\n  journal = {The Electronic Journal of Combinatorics},\n  volume = {19},\n  number = {3},\n  pages = {P33},\n  doi = {10.37236/2674}\n}\n",
+      "csl": {
+        "DOI": "10.37236/2674",
+        "author": [
+          {
+            "family": "Liu",
+            "given": "Lily L."
+          }
+        ],
+        "container-title": "The Electronic Journal of Combinatorics",
+        "issue": "3",
+        "issued": {
+          "date-parts": [
+            [
+              2012
+            ]
+          ]
+        },
+        "page": "P33",
+        "title": "Polynomials with real zeros and compatible sequences",
+        "type": "article-journal",
+        "volume": "19"
+      },
+      "label": "Liu12"
+    },
+    "LiuMao2026Hadamard": {
+      "bibtex": "@article{LiuMao2026Hadamard,\n  author = {Yanxin Liu and Jianxi Mao},\n  title = {Preservation of log-concavity under {H}adamard products},\n  year = {2026},\n  eprint = {2609.11589v1},\n  url = {https://arxiv.org/abs/2609.11589v1},\n  journal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/2609.11589v1",
+        "author": [
+          {
+            "family": "Liu",
+            "given": "Yanxin"
+          },
+          {
+            "family": "Mao",
+            "given": "Jianxi"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2026
+            ]
+          ]
+        },
+        "title": "Preservation of log-concavity under Hadamard products",
+        "type": "article-journal"
+      },
+      "label": "LM26"
+    },
+    "LiuWang2007": {
+      "bibtex": "@article{LiuWang2007,\n  doi = {10.1016/j.aam.2006.02.003},\n  url2 = {https://doi.org/10.1016/j.aam.2006.02.003},\n  year = {2007},\n  month = may,\n  publisher = {Elsevier {BV}},\n  volume = {38},\n  number = {4},\n  pages = {542--560},\n  author = {Lily L. Liu and Yi Wang},\n  title = {A unified approach to polynomial sequences with only real zeros},\n  journal = {Advances in Applied Mathematics}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.aam.2006.02.003",
+        "author": [
+          {
+            "family": "Liu",
+            "given": "Lily L."
+          },
+          {
+            "family": "Wang",
+            "given": "Yi"
+          }
+        ],
+        "container-title": "Advances in Applied Mathematics",
+        "issue": "4",
+        "issued": {
+          "date-parts": [
+            [
+              2007,
+              5
+            ]
+          ]
+        },
+        "page": "542-560",
+        "title": "A unified approach to polynomial sequences with only real zeros",
+        "type": "article-journal",
+        "volume": "38"
+      },
+      "label": "LW07"
+    },
+    "LiuZhang2026bx": {
+      "bibtex": "@article{LiuZhang2026bx,\n  author = {Feihu Liu and Zihao Zhang},\n  title = {Real-{R}ootedness and {G}amma-{P}ositivity for a {V}ariation of the {M}orris {C}onstant {T}erm},\n  year = {2026},\n  eprint = {2609.15201v3},\n  url = {https://arxiv.org/abs/2609.15201v3},\n  journal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/2609.15201v3",
+        "author": [
+          {
+            "family": "Liu",
+            "given": "Feihu"
+          },
+          {
+            "family": "Zhang",
+            "given": "Zihao"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2026
+            ]
+          ]
+        },
+        "title": "Real-Rootedness and Gamma-Positivity for a Variation of the Morris Constant Term",
+        "type": "article-journal"
+      },
+      "label": "LZ26"
+    },
+    "MaMaYeh2017x": {
+      "bibtex": "@article{MaMaYeh2017x,\n  author = {Jun Ma and Shi-Mei Ma and Yeong-Nan Yeh},\n  title = {Recurrence relations for binomial-{E}ulerian polynomials},\n  year = {2017},\n  eprint = {1711.09016v2},\n  url = {https://arxiv.org/abs/1711.09016v2},\n  journal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/1711.09016v2",
+        "author": [
+          {
+            "family": "Ma",
+            "given": "Jun"
+          },
+          {
+            "family": "Ma",
+            "given": "Shi-Mei"
+          },
+          {
+            "family": "Yeh",
+            "given": "Yeong-Nan"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2017
+            ]
+          ]
+        },
+        "title": "Recurrence relations for binomial-Eulerian polynomials",
+        "type": "article-journal"
+      },
+      "label": "MMY17"
+    },
+    "MaWang2008": {
+      "bibtex": "@article{MaWang2008,\n  doi = {10.37236/741},\n  url2 = {https://doi.org/10.37236/741},\n  year = {2008},\n  month = jan,\n  publisher = {The Electronic Journal of Combinatorics},\n  volume = {15},\n  number = {1},\n  author = {Shi-Mei Ma and Yi Wang},\n  title = {$q$-{E}ulerian Polynomials and Polynomials with Only Real Zeros},\n  pages = {Research Paper 17},\n  journal = {The Electronic Journal of Combinatorics}\n}\n",
+      "csl": {
+        "DOI": "10.37236/741",
+        "author": [
+          {
+            "family": "Ma",
+            "given": "Shi-Mei"
+          },
+          {
+            "family": "Wang",
+            "given": "Yi"
+          }
+        ],
+        "container-title": "The Electronic Journal of Combinatorics",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              2008,
+              1
+            ]
+          ]
+        },
+        "page": "Research Paper 17",
+        "title": "$q$-Eulerian polynomials and polynomials with only real zeros",
+        "type": "article-journal",
+        "volume": "15"
+      },
+      "label": "MW08"
+    },
+    "MaWang2026x": {
+      "bibtex": "@article{MaWang2026x,\n  author = {Shi-Mei Ma and Ming-Xin Wang},\n  title = {Binomial expansions of {J}acobi-{S}tirling numbers and real-rootedness of {J}acobi-{S}tirling descent polynomials},\n  year = {2026},\n  eprint = {2610.03111v1},\n  url = {https://arxiv.org/abs/2610.03111v1},\n  journal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/2610.03111v1",
+        "author": [
+          {
+            "family": "Ma",
+            "given": "Shi-Mei"
+          },
+          {
+            "family": "Wang",
+            "given": "Ming-Xin"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2026
+            ]
+          ]
+        },
+        "title": "Binomial expansions of Jacobi-Stirling numbers and real-rootedness of Jacobi-Stirling descent polynomials",
+        "type": "article-journal"
+      },
+      "label": "MW26"
+    },
+    "MantaciRakotondrajao2003": {
+      "bibtex": "@article{MantaciRakotondrajao2003,\n  doi = {10.1016/s0196-8858(02)00531-6},\n  url2 = {https://doi.org/10.1016/s0196-8858(02)00531-6},\n  year = {2003},\n  month = feb,\n  publisher = {Elsevier {BV}},\n  volume = {30},\n  number = {1-2},\n  pages = {177--188},\n  author = {Roberto Mantaci and Fanja Rakotondrajao},\n  title = {Exceedingly deranging!},\n  journal = {Advances in Applied Mathematics}\n}\n",
+      "csl": {
+        "DOI": "10.1016/s0196-8858(02)00531-6",
+        "author": [
+          {
+            "family": "Mantaci",
+            "given": "Roberto"
+          },
+          {
+            "family": "Rakotondrajao",
+            "given": "Fanja"
+          }
+        ],
+        "container-title": "Advances in Applied Mathematics",
+        "issue": "1-2",
+        "issued": {
+          "date-parts": [
+            [
+              2003,
+              2
+            ]
+          ]
+        },
+        "page": "177-188",
+        "title": "Exceedingly deranging!",
+        "type": "article-journal",
+        "volume": "30"
+      },
+      "label": "MR03"
+    },
+    "MaoWang2026x": {
+      "bibtex": "@article{MaoWang2026x,\n  author = {Jianxi Mao and Lijie Wang},\n  title = {The {N}arayana transformation},\n  year = {2026},\n  eprint = {2607.01572},\n  url = {https://arxiv.org/abs/2607.01572},\n  journal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/2607.01572",
+        "author": [
+          {
+            "family": "Mao",
+            "given": "Jianxi"
+          },
+          {
+            "family": "Wang",
+            "given": "Lijie"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2026
+            ]
+          ]
+        },
+        "title": "The Narayana transformation",
+        "type": "article-journal"
+      },
+      "label": "MW26"
+    },
+    "MarcusSpielmanSrivastava2015II": {
+      "bibtex": "@article{MarcusSpielmanSrivastava2015II,\n  author = {Adam Marcus and Daniel A. Spielman and Nikhil Srivastava},\n  title = {Interlacing families {II}: Mixed characteristic polynomials and the\n    {K}adison--{S}inger problem},\n  year = {2015},\n  journal = {Annals of Mathematics},\n  volume = {182},\n  number = {1},\n  pages = {327--350},\n  doi = {10.4007/annals.2015.182.1.8},\n  url = {https://doi.org/10.4007/annals.2015.182.1.8},\n  eprint = {1306.3969}\n}\n",
+      "csl": {
+        "DOI": "10.4007/annals.2015.182.1.8",
+        "URL": "https://arxiv.org/abs/1306.3969",
+        "author": [
+          {
+            "family": "Marcus",
+            "given": "Adam"
+          },
+          {
+            "family": "Spielman",
+            "given": "Daniel A."
+          },
+          {
+            "family": "Srivastava",
+            "given": "Nikhil"
+          }
+        ],
+        "container-title": "Annals of Mathematics",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              2015
+            ]
+          ]
+        },
+        "page": "327-350",
+        "title": "Interlacing families II: Mixed characteristic polynomials and the Kadison–Singer problem",
+        "type": "article-journal",
+        "volume": "182"
+      },
+      "label": "MSS15"
+    },
+    "MarcusSpielmanSrivastava2022": {
+      "bibtex": "@article{MarcusSpielmanSrivastava2022,\n  author = {Marcus, Adam W. and Spielman, Daniel A. and Srivastava, Nikhil},\n  title = {Finite free convolutions of polynomials},\n  year = {2022},\n  journal = {Probability Theory and Related Fields},\n  volume = {182},\n  number = {3-4},\n  pages = {807--848},\n  doi = {10.1007/s00440-021-01105-w},\n  url = {https://doi.org/10.1007/s00440-021-01105-w}\n}\n",
+      "csl": {
+        "DOI": "10.1007/s00440-021-01105-w",
+        "URL": "https://doi.org/10.1007/s00440-021-01105-w",
+        "author": [
+          {
+            "family": "Marcus",
+            "given": "Adam W."
+          },
+          {
+            "family": "Spielman",
+            "given": "Daniel A."
+          },
+          {
+            "family": "Srivastava",
+            "given": "Nikhil"
+          }
+        ],
+        "container-title": "Probability Theory and Related Fields",
+        "issue": "3-4",
+        "issued": {
+          "date-parts": [
+            [
+              2022
+            ]
+          ]
+        },
+        "page": "807-848",
+        "title": "Finite free convolutions of polynomials",
+        "type": "article-journal",
+        "volume": "182"
+      },
+      "label": "MSS22"
+    },
+    "MicchelliWilloughby1979": {
+      "bibtex": "@article{MicchelliWilloughby1979,\n  author = {Micchelli, Charles A. and Willoughby, R.A.},\n  title = {On functions which preserve the class of {S}tieltjes matrices},\n  year = {1979},\n  journal = {Linear Algebra and its Applications},\n  volume = {23},\n  pages = {141--156},\n  doi = {10.1016/0024-3795(79)90098-3},\n  url = {https://doi.org/10.1016/0024-3795(79)90098-3}\n}\n",
+      "csl": {
+        "DOI": "10.1016/0024-3795(79)90098-3",
+        "URL": "https://doi.org/10.1016/0024-3795(79)90098-3",
+        "author": [
+          {
+            "family": "Micchelli",
+            "given": "Charles A."
+          },
+          {
+            "family": "Willoughby",
+            "given": "R. A."
+          }
+        ],
+        "container-title": "Linear Algebra and its Applications",
+        "issued": {
+          "date-parts": [
+            [
+              1979
+            ]
+          ]
+        },
+        "page": "141-156",
+        "title": "On functions which preserve the class of Stieltjes matrices",
+        "type": "article-journal",
+        "volume": "23"
+      },
+      "label": "MW79"
+    },
+    "Mongelli2012": {
+      "bibtex": "@article{Mongelli2012,\n  author = {Mongelli, Pietro},\n  title = {Total positivity properties of {J}acobi–{S}tirling numbers},\n  year = {2012},\n  journal = {Advances in Applied Mathematics},\n  volume = {48},\n  number = {2},\n  pages = {354--364},\n  doi = {10.1016/j.aam.2011.06.008},\n  url = {https://doi.org/10.1016/j.aam.2011.06.008}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.aam.2011.06.008",
+        "URL": "https://doi.org/10.1016/j.aam.2011.06.008",
+        "author": [
+          {
+            "family": "Mongelli",
+            "given": "Pietro"
+          }
+        ],
+        "container-title": "Advances in Applied Mathematics",
+        "issue": "2",
+        "issued": {
+          "date-parts": [
+            [
+              2012
+            ]
+          ]
+        },
+        "page": "354-364",
+        "title": "Total positivity properties of Jacobi–Stirling numbers",
+        "type": "article-journal",
+        "volume": "48"
+      },
+      "label": "Mon12"
+    },
+    "Nijenhuis1976": {
+      "bibtex": "@article{Nijenhuis1976,\n  doi = {10.1016/0097-3165(76)90068-6},\n  url2 = {https://doi.org/10.1016/0097-3165(76)90068-6},\n  year = {1976},\n  month = sep,\n  publisher = {Elsevier {BV}},\n  volume = {21},\n  number = {2},\n  pages = {240--244},\n  author = {Albert Nijenhuis},\n  title = {On permanents and the zeros of rook polynomials},\n  journal = {Journal of Combinatorial Theory,  Series A}\n}\n",
+      "csl": {
+        "DOI": "10.1016/0097-3165(76)90068-6",
+        "author": [
+          {
+            "family": "Nijenhuis",
+            "given": "Albert"
+          }
+        ],
+        "container-title": "Journal of Combinatorial Theory, Series A",
+        "issue": "2",
+        "issued": {
+          "date-parts": [
+            [
+              1976,
+              9
+            ]
+          ]
+        },
+        "page": "240-244",
+        "title": "On permanents and the zeros of rook polynomials",
+        "type": "article-journal",
+        "volume": "21"
+      },
+      "label": "Nij76"
+    },
+    "NyandwiHaggKurujyibwamiUwimbabazi2026x": {
+      "bibtex": "@article{NyandwiHaggKurujyibwamiUwimbabazi2026x,\n  author = {Bosco Nyandwi and Christian Hägg and Celestin Kurujyibwami and Leon Fidele Ruganzu Uwimbabazi},\n  title = {Voronoi limit measures for iterates of constant-coefficient differential operators on rational functions with simple poles},\n  year = {2026},\n  eprint = {2604.05189v1},\n  url = {https://arxiv.org/abs/2604.05189v1},\n  journal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/2604.05189v1",
+        "author": [
+          {
+            "family": "Nyandwi",
+            "given": "Bosco"
+          },
+          {
+            "family": "Hägg",
+            "given": "Christian"
+          },
+          {
+            "family": "Kurujyibwami",
+            "given": "Celestin"
+          },
+          {
+            "family": "Uwimbabazi",
+            "given": "Leon Fidele Ruganzu"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2026
+            ]
+          ]
+        },
+        "title": "Voronoi limit measures for iterates of constant-coefficient differential operators on rational functions with simple poles",
+        "type": "article-journal"
+      },
+      "label": "NHKU26"
+    },
+    "Obreschkoff1963": {
+      "bibtex": "@book{Obreschkoff1963,\n  author = {Obreschkoff, Nikola},\n  title = {Verteilung und Berechnung der Nullstellen reeller Polynome},\n  series = {Hochschulb{\\\"u}cher f{\\\"u}r Mathematik},\n  volume = {55},\n  publisher = {VEB Deutscher Verlag der Wissenschaften},\n  address = {Berlin},\n  year = {1963}\n}\n",
+      "csl": {
+        "author": [
+          {
+            "family": "Obreschkoff",
+            "given": "Nikola"
+          }
+        ],
+        "issued": {
+          "date-parts": [
+            [
+              1963
+            ]
+          ]
+        },
+        "title": "Verteilung und berechnung der nullstellen reeller polynome",
+        "type": "book",
+        "volume": "55"
+      },
+      "label": "Obr63"
+    },
+    "Perron1907": {
+      "bibtex": "@article{Perron1907,\n  author = {Perron, Oskar},\n  title = {Zur {T}heorie der {M}atrices},\n  year = {1907},\n  journal = {Mathematische Annalen},\n  volume = {64},\n  number = {2},\n  pages = {248--263},\n  doi = {10.1007/BF01449896},\n  url = {https://doi.org/10.1007/BF01449896}\n}\n",
+      "csl": {
+        "DOI": "10.1007/BF01449896",
+        "URL": "https://doi.org/10.1007/BF01449896",
+        "author": [
+          {
+            "family": "Perron",
+            "given": "Oskar"
+          }
+        ],
+        "container-title": "Mathematische Annalen",
+        "issue": "2",
+        "issued": {
+          "date-parts": [
+            [
+              1907
+            ]
+          ]
+        },
+        "page": "248-263",
+        "title": "Zur Theorie der Matrices",
+        "type": "article-journal",
+        "volume": "64"
+      },
+      "label": "Per07"
+    },
+    "Petersen2015EulerianNumbers": {
+      "bibtex": "@book{Petersen2015EulerianNumbers,\n  author = {T. Kyle Petersen},\n  title = {Eulerian Numbers},\n  series = {Birkhäuser Advanced Texts Basler Lehrbücher},\n  publisher = {Birkhäuser},\n  address = {New York},\n  year = {2015},\n  isbn = {978-1-4939-3091-3},\n  doi = {10.1007/978-1-4939-3091-3},\n  url = {https://doi.org/10.1007/978-1-4939-3091-3}\n}\n",
+      "csl": {
+        "DOI": "10.1007/978-1-4939-3091-3",
+        "URL": "https://doi.org/10.1007/978-1-4939-3091-3",
+        "author": [
+          {
+            "family": "Petersen",
+            "given": "T. Kyle"
+          }
+        ],
+        "issued": {
+          "date-parts": [
+            [
+              2015
+            ]
+          ]
+        },
+        "title": "Eulerian numbers",
+        "type": "book"
+      },
+      "label": "Pet15"
+    },
+    "RahmanSchmeisser2002": {
+      "bibtex": "@book{RahmanSchmeisser2002,\n  author    = {Rahman, Q. I. and Schmeisser, G.},\n  title     = {Analytic Theory of Polynomials},\n  publisher = {Oxford University Press},\n  series    = {London Mathematical Society Monographs},\n  year      = {2002}\n}\n",
+      "csl": {
+        "author": [
+          {
+            "family": "Rahman",
+            "given": "Q. I."
+          },
+          {
+            "family": "Schmeisser",
+            "given": "G."
+          }
+        ],
+        "issued": {
+          "date-parts": [
+            [
+              2002
+            ]
+          ]
+        },
+        "title": "Analytic theory of polynomials",
+        "type": "book"
+      },
+      "label": "RS02"
+    },
+    "Ruelle2010": {
+      "bibtex": "@article{Ruelle2010,\n  author = {Ruelle, David},\n  title = {Characterization of {L}ee-{Y}ang polynomials},\n  year = {2010},\n  journal = {Annals of Mathematics},\n  volume = {171},\n  number = {1},\n  pages = {589--603},\n  doi = {10.4007/annals.2010.171.589},\n  url = {https://doi.org/10.4007/annals.2010.171.589}\n}\n",
+      "csl": {
+        "DOI": "10.4007/annals.2010.171.589",
+        "URL": "https://doi.org/10.4007/annals.2010.171.589",
+        "author": [
+          {
+            "family": "Ruelle",
+            "given": "David"
+          }
+        ],
+        "container-title": "Annals of Mathematics",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              2010
+            ]
+          ]
+        },
+        "page": "589-603",
+        "title": "Characterization of Lee-Yang polynomials",
+        "type": "article-journal",
+        "volume": "171"
+      },
+      "label": "Rue10"
+    },
+    "Schur1917": {
+      "bibtex": "@article{Schur1917,\n  author = {Schur, J.},\n  title = {Über {P}otenzreihen, die im {I}nnern des {E}inheitskreises beschränkt sind.},\n  year = {1917},\n  journal = {crll},\n  volume = {1917},\n  number = {147},\n  pages = {205--232},\n  doi = {10.1515/crll.1917.147.205},\n  url = {https://doi.org/10.1515/crll.1917.147.205}\n}\n",
+      "csl": {
+        "DOI": "10.1515/crll.1917.147.205",
+        "URL": "https://doi.org/10.1515/crll.1917.147.205",
+        "author": [
+          {
+            "family": "Schur",
+            "given": "J."
+          }
+        ],
+        "container-title": "crll",
+        "issue": "147",
+        "issued": {
+          "date-parts": [
+            [
+              1917
+            ]
+          ]
+        },
+        "page": "205-232",
+        "title": "Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind.",
+        "type": "article-journal",
+        "volume": "1917"
+      },
+      "label": "Sch17"
+    },
+    "SchurPolya1914": {
+      "bibtex": "@article{SchurPolya1914,\n  title={{\\\"U}ber zwei {A}rten von {F}aktorenfolgen in der {T}heorie der algebraischen {G}leichungen},\n  author={Schur, J and Polya, G},\n  year={1914},\n  journal={J. Reine Angew. Math.},\n  publisher={Walter de Gruyter, Berlin/New York Berlin, New York}\n}\n",
+      "csl": {
+        "author": [
+          {
+            "family": "Schur",
+            "given": "J"
+          },
+          {
+            "family": "Polya",
+            "given": "G"
+          }
+        ],
+        "container-title": "J. Reine Angew. Math.",
+        "issued": {
+          "date-parts": [
+            [
+              1914
+            ]
+          ]
+        },
+        "title": "Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen",
+        "type": "article-journal"
+      },
+      "label": "SP14"
+    },
+    "Sylvester1853": {
+      "bibtex": "@article{Sylvester1853,\n  author = {Sylvester, James Joseph},\n  title = {{XVIII}. {O}n a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of {S}turm's functions, and that of the greatest algebraical common measure},\n  year = {1853},\n  journal = {Philosophical Transactions of the Royal Society of London},\n  number = {143},\n  pages = {407--548},\n  doi = {10.1098/rstl.1853.0018},\n  url = {https://doi.org/10.1098/rstl.1853.0018}\n}\n",
+      "csl": {
+        "DOI": "10.1098/rstl.1853.0018",
+        "URL": "https://doi.org/10.1098/rstl.1853.0018",
+        "author": [
+          {
+            "family": "Sylvester",
+            "given": "James Joseph"
+          }
+        ],
+        "container-title": "Philosophical Transactions of the Royal Society of London",
+        "issue": "143",
+        "issued": {
+          "date-parts": [
+            [
+              1853
+            ]
+          ]
+        },
+        "page": "407-548",
+        "title": "XVIII. On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm’s functions, and that of the greatest algebraical common measure",
+        "type": "article-journal"
+      },
+      "label": "Syl53"
+    },
+    "Szego1922": {
+      "bibtex": "@article{Szego1922,\n  doi     = {10.1007/BF01485280},\n  url2    = {https://doi.org/10.1007/BF01485280},\n  author  = {G. Szeg{\\\"{o}}},\n  title   = {Bemerkungen zu einem {S}atz von {J}. {H}. {G}race {\\\"{u}}ber die {W}urzeln algebraischer {G}leichungen},\n  journal = {Mathematische Zeitschrift},\n  volume  = {13},\n  pages   = {28--55},\n  year    = {1922}\n}\n",
+      "csl": {
+        "DOI": "10.1007/BF01485280",
+        "author": [
+          {
+            "family": "Szegö",
+            "given": "G."
+          }
+        ],
+        "container-title": "Mathematische Zeitschrift",
+        "issued": {
+          "date-parts": [
+            [
+              1922
+            ]
+          ]
+        },
+        "page": "28-55",
+        "title": "Bemerkungen zu einem Satz von J. H. Grace über die Wurzeln algebraischer Gleichungen",
+        "type": "article-journal",
+        "volume": "13"
+      },
+      "label": "Sze22"
+    },
+    "Szego1939": {
+      "bibtex": "@book{Szego1939,\n  author = {Szegő, G.},\n  title = {Orthogonal {P}olynomials},\n  year = {1939},\n  series = {Colloquium Publications},\n  publisher = {American Mathematical Society},\n  isbn = {9781470431716},\n  doi = {10.1090/coll/023},\n  url = {https://doi.org/10.1090/coll/023}\n}\n",
+      "csl": {
+        "DOI": "10.1090/coll/023",
+        "URL": "https://doi.org/10.1090/coll/023",
+        "author": [
+          {
+            "family": "Szegő",
+            "given": "G."
+          }
+        ],
+        "issued": {
+          "date-parts": [
+            [
+              1939
+            ]
+          ]
+        },
+        "title": "Orthogonal Polynomials",
+        "type": "book"
+      },
+      "label": "Sze39"
+    },
+    "Wagner1992": {
+      "bibtex": "@article{Wagner1992,\n  doi = {10.1016/0022-247x(92)90261-b},\n  url2 = {https://doi.org/10.1016/0022-247x(92)90261-b},\n  year = {1992},\n  month = jan,\n  publisher = {Elsevier {BV}},\n  volume = {163},\n  number = {2},\n  pages = {459--483},\n  author = {David G Wagner},\n  title = {Total positivity of {H}adamard products},\n  journal = {Journal of Mathematical Analysis and Applications}\n}\n",
+      "csl": {
+        "DOI": "10.1016/0022-247x(92)90261-b",
+        "author": [
+          {
+            "family": "Wagner",
+            "given": "David G"
+          }
+        ],
+        "container-title": "Journal of Mathematical Analysis and Applications",
+        "issue": "2",
+        "issued": {
+          "date-parts": [
+            [
+              1992,
+              1
+            ]
+          ]
+        },
+        "page": "459-483",
+        "title": "Total positivity of Hadamard products",
+        "type": "article-journal",
+        "volume": "163"
+      },
+      "label": "Wag92"
+    },
+    "Walsh1922": {
+      "bibtex": "@article{Walsh1922,\n  doi     = {10.1090/S0002-9947-1922-1501220-0},\n  url2    = {https://doi.org/10.1090/S0002-9947-1922-1501220-0},\n  author  = {J. L. Walsh},\n  title   = {On the location of the roots of certain types of polynomials},\n  journal = {Transactions of the American Mathematical Society},\n  volume  = {24},\n  number  = {3},\n  pages   = {163--180},\n  year    = {1922}\n}\n",
+      "csl": {
+        "DOI": "10.1090/S0002-9947-1922-1501220-0",
+        "author": [
+          {
+            "family": "Walsh",
+            "given": "J. L."
+          }
+        ],
+        "container-title": "Transactions of the American Mathematical Society",
+        "issue": "3",
+        "issued": {
+          "date-parts": [
+            [
+              1922
+            ]
+          ]
+        },
+        "page": "163-180",
+        "title": "On the location of the roots of certain types of polynomials",
+        "type": "article-journal",
+        "volume": "24"
+      },
+      "label": "Wal22"
+    },
+    "Wang2014": {
+      "bibtex": "@article{Wang2014,\n  title = {On colored set partitions of type $B_n$},\n  volume = {12},\n  ISSN = {2391-5455},\n  url = {http://dx.doi.org/10.2478/s11533-014-0419-9},\n  DOI = {10.2478/s11533-014-0419-9},\n  number = {9},\n  journal = {Open Mathematics},\n  publisher = {Walter de Gruyter GmbH},\n  author = {Wang,  David},\n  year = {2014},\n  month = jan\n}\n",
+      "csl": {
+        "DOI": "10.2478/s11533-014-0419-9",
+        "URL": "http://dx.doi.org/10.2478/s11533-014-0419-9",
+        "author": [
+          {
+            "family": "Wang",
+            "given": "David"
+          }
+        ],
+        "container-title": "Open Mathematics",
+        "issue": "9",
+        "issued": {
+          "date-parts": [
+            [
+              2014,
+              1
+            ]
+          ]
+        },
+        "title": "On colored set partitions of type $B_n$",
+        "type": "article-journal",
+        "volume": "12"
+      },
+      "label": "Wan14"
+    },
+    "WangYeh2005": {
+      "bibtex": "@article{WangYeh2005,\n  author = {Yi Wang and Yeong-Nan Yeh},\n  title = {Polynomials with real zeros and {P}{\\'o}lya frequency sequences},\n  year = {2005},\n  journal = {Journal of Combinatorial Theory, Series A},\n  volume = {109},\n  number = {1},\n  pages = {63--74},\n  doi = {10.1016/j.jcta.2004.07.008},\n  url = {https://doi.org/10.1016/j.jcta.2004.07.008},\n  eprint = {math/0611825}\n}\n",
+      "csl": {
+        "DOI": "10.1016/j.jcta.2004.07.008",
+        "URL": "https://arxiv.org/abs/math/0611825",
+        "author": [
+          {
+            "family": "Wang",
+            "given": "Yi"
+          },
+          {
+            "family": "Yeh",
+            "given": "Yeong-Nan"
+          }
+        ],
+        "container-title": "Journal of Combinatorial Theory, Series A",
+        "issue": "1",
+        "issued": {
+          "date-parts": [
+            [
+              2005
+            ]
+          ]
+        },
+        "page": "63-74",
+        "title": "Polynomials with real zeros and Pólya frequency sequences",
+        "type": "article-journal",
+        "volume": "109"
+      },
+      "label": "WY05"
+    },
+    "Xiao2026x": {
+      "bibtex": "@article{Xiao2026x,\n  author = {Qiqi Xiao},\n  title = {The real-rootedness of the toric $g$-contribution polynomials},\n  year = {2026},\n  eprint = {2609.01086v1},\n  url = {https://arxiv.org/abs/2609.01086v1},\n  journal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/2609.01086v1",
+        "author": [
+          {
+            "family": "Xiao",
+            "given": "Qiqi"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2026
+            ]
+          ]
+        },
+        "title": "The real-rootedness of the toric $g$-contribution polynomials",
+        "type": "article-journal"
+      },
+      "label": "Xia26"
+    },
+    "XieZhang2026InfiniteLogConcavity": {
+      "bibtex": "@article{XieZhang2026InfiniteLogConcavity,\n  author = {Matthew H. Y. Xie and Philip B. Zhang},\n  title = {Infinite log-concavity of the {B}oros--{M}oll sequences},\n  year = {2026},\n  eprint = {2609.20653v1},\n  url = {https://arxiv.org/abs/2609.20653v1},\n  journal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/2609.20653v1",
+        "author": [
+          {
+            "family": "Xie",
+            "given": "Matthew H. Y."
+          },
+          {
+            "family": "Zhang",
+            "given": "Philip B."
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2026
+            ]
+          ]
+        },
+        "title": "Infinite log-concavity of the Boros–Moll sequences",
+        "type": "article-journal"
+      },
+      "label": "XZ26"
+    },
+    "Zeng2026x": {
+      "bibtex": "@article{Zeng2026x,\n  author = {Jiang Zeng},\n  title = {Continuous {N}arayana {I}nterpolations: {H}ankel {T}otal {P}ositivity and {L}og-{C}onvexity {P}reservation},\n  year = {2026},\n  eprint = {2610.19548v3},\n  url = {https://arxiv.org/abs/2610.19548v3},\n  journal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/2610.19548v3",
+        "author": [
+          {
+            "family": "Zeng",
+            "given": "Jiang"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2026
+            ]
+          ]
+        },
+        "title": "Continuous Narayana Interpolations: Hankel Total Positivity and Log-Convexity Preservation",
+        "type": "article-journal"
+      },
+      "label": "Zen26"
+    },
+    "Zhang2026SeparableInterlacing": {
+      "bibtex": "@unpublished{Zhang2026SeparableInterlacing,\n  author = {Zhang, Zhanhe},\n  title = {Strict Interlacing of the Descent Polynomials of Separable Permutations},\n  date = {2026-09-10},\n  year = {2026},\n  note = {SSRN preprint 7510941, posted September 25, 2026},\n  doi = {10.2139/ssrn.7510941},\n  url = {https://ssrn.com/abstract=7510941}\n}\n",
+      "csl": {
+        "DOI": "10.2139/ssrn.7510941",
+        "URL": "https://ssrn.com/abstract=7510941",
+        "author": [
+          {
+            "family": "Zhang",
+            "given": "Zhanhe"
+          }
+        ],
+        "issued": {
+          "date-parts": [
+            [
+              2026,
+              9,
+              10
+            ]
+          ]
+        },
+        "title": "Strict interlacing of the descent polynomials of separable permutations",
+        "type": "manuscript"
+      },
+      "label": "Zha26"
+    },
+    "ZhangZhao2026x": {
+      "bibtex": "@article{ZhangZhao2026x,\n  author = {Philip B. Zhang and Tongyuan Zhao},\n  title = {Zeros and interlacing for multiset {E}ulerian--{N}arayana polynomials},\n  year = {2026},\n  eprint = {2610.00966v1},\n  url = {https://arxiv.org/abs/2610.00966v1},\n  journal = {arXiv e-prints}\n}\n",
+      "csl": {
+        "URL": "https://arxiv.org/abs/2610.00966v1",
+        "author": [
+          {
+            "family": "Zhang",
+            "given": "Philip B."
+          },
+          {
+            "family": "Zhao",
+            "given": "Tongyuan"
+          }
+        ],
+        "container-title": "arXiv e-prints",
+        "issued": {
+          "date-parts": [
+            [
+              2026
+            ]
+          ]
+        },
+        "title": "Zeros and interlacing for multiset Eulerian–Narayana polynomials",
+        "type": "article-journal"
+      },
+      "label": "ZZ26"
+    }
+  },
+  "schema_version": 1
+}
diff --git a/website/symcat/labels.json b/website/symcat/labels.json
new file mode 100644
index 000000000..6bc4e0b25
--- /dev/null
+++ b/website/symcat/labels.json
@@ -0,0 +1 @@
+{"403":"403.htm#403","404":"404.htm#404","6vertexModel":"yang-baxter.htm#6vertexModel","A189912":"various-research.htm#A189912","A358628Columns":"various-research.htm#A358628Columns","KatalanSymmetric":"kschur.htm#KatalanSymmetric","KazhdanLusztig":"matroids.htm#KazhdanLusztig","LGVLatticePathsInSkewShape":"lgv-lemma.htm#LGVLatticePathsInSkewShape","LGVplanePartitions":"lgv-lemma.htm#LGVplanePartitions","LGVssyt":"lgv-lemma.htm#LGVssyt","LLT":"llt.htm#LLT","MConvexity":"newtonPolytopes.htm#MConvexity","NCSYMBases":"ncSchur.htm#NCSYMBases","NCSymCompleteHomogeneousBasis":"ncSchur.htm#NCSymCompleteHomogeneousBasis","NCSymElementaryBasis":"ncSchur.htm#NCSymElementaryBasis","NCSymHopfAlgebra":"ncSchur.htm#NCSymHopfAlgebra","NCSymMonomialBasis":"ncSchur.htm#NCSymMonomialBasis","NCSymPermutationMap":"ncSchur.htm#NCSymPermutationMap","NCSymPowerSumBasis":"ncSchur.htm#NCSymPowerSumBasis","NCSymStandardBases":"ncSchur.htm#NCSymStandardBases","RSKstatistics":"rsk.htm#RSKstatistics","RSKvsDualRSK":"tableauOperators.htm#RSKvsDualRSK","SNP":"newtonPolytopes.htm#SNP","SchurNoncommuting":"ncSchur.htm#SchurNoncommuting","SnRepresentationTheory":"representationTheory.htm#SnRepresentationTheory","YangBaxter":"latticeModel.htm#YangBaxter","aCompatibleSequence":"schubert.htm#aCompatibleSequence","abelianPGroupCotype":"hallLittlewood.htm#abelianPGroupCotype","abelianUnitIntervalGraphs":"chromaticQuasisymmetric.htm#abelianUnitIntervalGraphs","about":"about.htm#about","aboutContent":"about.htm#aboutContent","aboutContributions":"about.htm#aboutContributions","aboutWebpage":"about.htm#aboutWebpage","acyclicOrientation":"chromaticQuasisymmetric.htm#acyclicOrientation","adamsOperator":"plethysm.htm#adamsOperator","adamsOperatorSymmetricFunctions":"symmetricFunctionsSpace.htm#adamsOperatorSymmetricFunctions","affineEvacuation":"tableauOperators.htm#affineEvacuation","affineInvolutionStanleySymmetricFunction":"stanleySymmetric.htm#affineInvolutionStanleySymmetricFunction","affineJackPolynomial":"jack.htm#affineJackPolynomial","affineLittlewoodIdentity":"cylindricSchur.htm#affineLittlewoodIdentity","affineProjectiveVarieties":"varieties.htm#affineProjectiveVarieties","affineSchubertPolynomial":"schubertVariations.htm#affineSchubertPolynomial","affineSchurFunction":"stanleySymmetric.htm#affineSchurFunction","affineStanleySymmetricFunction":"stanleySymmetric.htm#affineStanleySymmetricFunction","affineVarietyDefinition":"varieties.htm#affineVarietyDefinition","aissenSchoenbergWhitney":"polyaFrequency.htm#aissenSchoenbergWhitney","algebraicStabilitySymbols":"stablePolynomials.htm#algebraicStabilitySymbols","algebraicSymbol":"stablePolynomials.htm#algebraicSymbol","algebras":"algebras-and-rings.htm#algebras","almkvistPartitionPolynomials":"q-analogs.htm#almkvistPartitionPolynomials","almostSymmetricSchur":"key.htm#almostSymmetricSchur","alphaChromaticSymmetric":"chromaticQuasisymmetric.htm#alphaChromaticSymmetric","alphaChromaticSymmetricFunction":"chromaticQuasisymmetric.htm#alphaChromaticSymmetricFunction","alternatesLeftOf":"realRootedInterlacing.htm#alternatesLeftOf","alternatingGammaPositivePolynomial":"gammaPositivity.htm#alternatingGammaPositivePolynomial","alternatingGammaPositivity":"gammaPositivity.htm#alternatingGammaPositivity","alternatingRunPolynomial":"realRootedWords.htm#alternatingRunPolynomial","alternatingSignMatrix":"gtpatterns.htm#alternatingSignMatrix","alternatingSignMatrixDefinition":"gtpatterns.htm#alternatingSignMatrixDefinition","antichain":"posets.htm#antichain","antichainBoxStability":"realRooted.htm#antichainBoxStability","aoMacdonald":"touchardRiordan.htm#aoMacdonald","aoMaps":"touchardRiordan.htm#aoMaps","aoOnUnitIntervalGraphs":"touchardRiordan.htm#aoOnUnitIntervalGraphs","aoOther":"touchardRiordan.htm#aoOther","aoPBWBasis":"touchardRiordan.htm#aoPBWBasis","aoPerfectMatchings":"touchardRiordan.htm#aoPerfectMatchings","aoRooks":"touchardRiordan.htm#aoRooks","apolarPolynomial":"stablePolynomials.htm#apolarPolynomial","areaSequence":"chromaticQuasisymmetric.htm#areaSequence","arrowPatterns":"permutationPatterns.htm#arrowPatterns","artinSymmetricFunction":"standardSymmetricFunctions.htm#artinSymmetricFunction","associahedron":"polytopes.htm#associahedron","associahedronDefinition":"polytopes.htm#associahedronDefinition","atom":"key.htm#atom","atomCauchyKernel":"key.htm#atomCauchyKernel","atomsTypeBC":"key.htm#atomsTypeBC","attackingBoxes":"macdonaldEperm.htm#attackingBoxes","augmentedFilling":"macdonaldEperm.htm#augmentedFilling","augmentedMonomial":"standardSymmetricFunctions.htm#augmentedMonomial","backStableCHM":"backStableSchubert.htm#backStableCHM","backStableDynkinReversal":"backStableSchubert.htm#backStableDynkinReversal","backStableSEM":"backStableSchubert.htm#backStableSEM","backStableSEMStabilization":"backStableSchubert.htm#backStableSEMStabilization","backStableSchubert":"backStableSchubert.htm#backStableSchubert","backStableSchubertPolynomial":"backStableSchubert.htm#backStableSchubertPolynomial","backStableSchubertRing":"backStableSchubert.htm#backStableSchubertRing","backStableShiftedSpecialization":"backStableSchubert.htm#backStableShiftedSpecialization","backStableStandardMonomials":"backStableSchubert.htm#backStableStandardMonomials","backStableStanleyOrbitSums":"backStableSchubert.htm#backStableStanleyOrbitSums","backwardExtSchur":"qsymSchur.htm#backwardExtSchur","backwardExtendedSchurFunction":"qsymSchur.htm#backwardExtendedSchurFunction","ballotWord":"preliminaries.htm#ballotWord","basement":"macdonaldEperm.htm#basement","basicHypergeometric":"q-analogs.htm#basicHypergeometric","basisExchangeAxiom":"matroids.htm#basisExchangeAxiom","basisGeneratingPolynomial":"matroids.htm#basisGeneratingPolynomial","baxterHoggattSturm":"realRootedInterlacing.htm#baxterHoggattSturm","benderKnuth":"tableauOperators.htm#benderKnuth","benderKnuthInvolution":"tableauOperators.htm#benderKnuthInvolution","berahaKahaneWeiss":"transfer-matrix.htm#berahaKahaneWeiss","bereleInsertion":"rsk.htm#bereleInsertion","bereleRSK":"rsk.htm#bereleRSK","berensteinZelevinskyPolytope":"gtpatterns.htm#berensteinZelevinskyPolytope","bergeronGarsiaOperator":"shuffleAndDelta.htm#bergeronGarsiaOperator","bernsteinOperator":"symmetricFunctionsSpace.htm#bernsteinOperator","betaDividedDifferenceOperator":"lascoux.htm#betaDividedDifferenceOperator","bezoutMatrix":"realRootedInterlacing.htm#bezoutMatrix","bezoutMatrixDefinition":"realRootedInterlacing.htm#bezoutMatrixDefinition","biEulerianPolynomial":"polytopes.htm#biEulerianPolynomial","bialternantFormula":"grothendieck.htm#bialternantFormula","bigDescent":"realRootedWords.htm#bigDescent","bigSchur":"superSymmetricSchur.htm#bigSchur","bigSchurFunction":"superSymmetricSchur.htm#bigSchurFunction","binomialEulerianPolynomial":"realRootedWords.htm#binomialEulerianPolynomial","binomialSEulerianPolynomial":"realRootedWords.htm#binomialSEulerianPolynomial","birkhoffPolytope":"polytopes.htm#birkhoffPolytope","birkhoffPolytopeAsGelfandTsetlin":"gtpatterns.htm#birkhoffPolytopeAsGelfandTsetlin","birkhoffPolytopeDefinition":"polytopes.htm#birkhoffPolytopeDefinition","bisymmetricMacdonaldPolynomial":"macdonaldP.htm#bisymmetricMacdonaldPolynomial","bivariateStableDeterminantalRepresentation":"stablePolynomials.htm#bivariateStableDeterminantalRepresentation","biword":"rsk.htm#biword","bondLattice":"weightedBondSymmetric.htm#bondLattice","booleanLattice":"posets.htm#booleanLattice","booleanNarayanaNumberDefinition":"realRootedCatalan.htm#booleanNarayanaNumberDefinition","booleanProduct":"booleanProductPolynomials.htm#booleanProduct","booleanProductBivariate":"booleanProductPolynomials.htm#booleanProductBivariate","booleanProductSchurPositivity":"booleanProductPolynomials.htm#booleanProductSchurPositivity","borceaBrandenFiniteSymbol":"stablePolynomials.htm#borceaBrandenFiniteSymbol","borceaBrandenSymbols":"stablePolynomials.htm#borceaBrandenSymbols","borceaBrandenTranscendentalSymbol":"stablePolynomials.htm#borceaBrandenTranscendentalSymbol","borderStrip":"preliminaries.htm#borderStrip","borderStripAbacus":"borderStripTableaux.htm#borderStripAbacus","borderStripHookFormulas":"borderStripTableaux.htm#borderStripHookFormulas","borderStripTableaux":"borderStripTableaux.htm#borderStripTableaux","borderStripTableauxDefinition":"borderStripTableaux.htm#borderStripTableauxDefinition","borelSubgroup":"varieties.htm#borelSubgroup","bosonicMacdonaldPolynomial":"macdonaldP.htm#bosonicMacdonaldPolynomial","bosonicPlethysmCluster":"plethysm.htm#bosonicPlethysmCluster","bottomSchurFunction":"schurMisc.htm#bottomSchurFunction","boundaryMeasurementMatrix":"positroids.htm#boundaryMeasurementMatrix","boundedRecurrenceSpaces":"polynomialRecurrences.htm#boundedRecurrenceSpaces","braidRelation":"permutations.htm#braidRelation","braidRelations":"permutations.htm#braidRelations","brentiEhrhartCriterion":"realRootedTableaux.htm#brentiEhrhartCriterion","brentiTnnPathMatrix":"polyaFrequency.htm#brentiTnnPathMatrix","brickTabloid":"standardSymmetricFunctions.htm#brickTabloid","bruhatOrder":"permutations.htm#bruhatOrder","bruhatOrderDefinition":"permutations.htm#bruhatOrderDefinition","bumplessPipeDream":"schubert.htm#bumplessPipeDream","burgeWords":"rsk.htm#burgeWords","bzPatternDefinition":"hivePolytopes.htm#bzPatternDefinition","bzPatterns":"littlewoodRichardson.htm#bzPatterns","calogeroSutherlandOperator":"jack.htm#calogeroSutherlandOperator","canonPermutation":"realRootedWords.htm#canonPermutation","catabolism":"kostkaFoulkes.htm#catabolism","catalanSymmetric":"kschur.htm#catalanSymmetric","catalanSymmetricDefinition":"kschur.htm#catalanSymmetricDefinition","catalanSymmetricFunctionDefinition":"kschur.htm#catalanSymmetricFunctionDefinition","cauchyBialternantFormula":"schur.htm#cauchyBialternantFormula","cauchyIdentity":"schur.htm#cauchyIdentity","cauchyInterlacingTheorem":"realRootedInterlacing.htm#cauchyInterlacingTheorem","cdIndex":"gammaPositivity.htm#cdIndex","centeredChromaticSymmetricFunction":"chromaticQuasisymmetric.htm#centeredChromaticSymmetricFunction","chainPolynomial":"posets.htm#chainPolynomial","chainPolytope":"polytopes.htm#chainPolytope","chainPolytopeDefinition":"polytopes.htm#chainPolytopeDefinition","chainPoset":"posets.htm#chainPoset","charSymFuncProps":"characterSymmetricFunctions.htm#charSymFuncProps","characterHookFormulaCase":"murnaghanNakayama.htm#characterHookFormulaCase","characterSymmetricFunctions":"characterSymmetricFunctions.htm#characterSymmetricFunctions","chargeProperties":"kostkaFoulkes.htm#chargeProperties","chargeStatistic":"kostkaFoulkes.htm#chargeStatistic","chernClassMNRule":"murnaghanNakayama.htm#chernClassMNRule","chiFormulas":"murnaghanNakayama.htm#chiFormulas","chordalGraph":"realRootedGraphs.htm#chordalGraph","chowPolynomialDefinition":"realRootedGraphs.htm#chowPolynomialDefinition","chromaticDeletionContraction":"chromaticQuasisymmetric.htm#chromaticDeletionContraction","chromaticEExpansionChronology":"chromaticEexpansion.htm#chromaticEExpansionChronology","chromaticEExpansionCoefficientTests":"chromaticEexpansion.htm#chromaticEExpansionCoefficientTests","chromaticEExpansionConjectures":"chromaticEexpansion.htm#chromaticEExpansionConjectures","chromaticEExpansionDefinition":"chromaticEexpansion.htm#chromaticEExpansionDefinition","chromaticEExpansionMethods":"chromaticEexpansion.htm#chromaticEExpansionMethods","chromaticEExpansionNegative":"chromaticEexpansion.htm#chromaticEExpansionNegative","chromaticEExpansionOrdinaryFamilies":"chromaticEexpansion.htm#chromaticEExpansionOrdinaryFamilies","chromaticEExpansionQRefined":"chromaticEexpansion.htm#chromaticEExpansionQRefined","chromaticEPositive":"chromaticEexpansion.htm#chromaticEPositive","chromaticEShareshianWachsConjecture":"chromaticEexpansion.htm#chromaticEShareshianWachsConjecture","chromaticEStarObstruction":"chromaticEexpansion.htm#chromaticEStarObstruction","chromaticEUnimodalityConjecture":"chromaticEexpansion.htm#chromaticEUnimodalityConjecture","chromaticEexpansion":"chromaticEexpansion.htm#chromaticEexpansion","chromaticMultisymmetricFunction":"chromaticQuasisymmetric.htm#chromaticMultisymmetricFunction","chromaticPolynomial":"chromaticQuasisymmetric.htm#chromaticPolynomial","chromaticProductsOfChains":"chromaticQuasisymmetric.htm#chromaticProductsOfChains","chromaticQuasisymmetric":"chromaticQuasisymmetric.htm#chromaticQuasisymmetric","chromaticQuasisymmetricClawFree":"chromaticQuasisymmetric.htm#chromaticQuasisymmetricClawFree","chromaticQuasisymmetricDefinition":"chromaticQuasisymmetric.htm#chromaticQuasisymmetricDefinition","chromaticQuasisymmetricElementaryExpansion":"chromaticQuasisymmetric.htm#chromaticQuasisymmetricElementaryExpansion","chromaticQuasisymmetricFunction":"chromaticQuasisymmetric.htm#chromaticQuasisymmetricFunction","chromaticQuasisymmetricGeneralGraphs":"chromaticQuasisymmetric.htm#chromaticQuasisymmetricGeneralGraphs","chromaticQuasisymmetricGesselExpansion":"chromaticQuasisymmetric.htm#chromaticQuasisymmetricGesselExpansion","chromaticQuasisymmetricHessenberg":"chromaticQuasisymmetric.htm#chromaticQuasisymmetricHessenberg","chromaticQuasisymmetricHistory":"chromaticQuasisymmetric.htm#chromaticQuasisymmetricHistory","chromaticQuasisymmetricLLT":"chromaticQuasisymmetric.htm#chromaticQuasisymmetricLLT","chromaticQuasisymmetricMisc":"chromaticQuasisymmetric.htm#chromaticQuasisymmetricMisc","chromaticQuasisymmetricMomentGraph":"chromaticQuasisymmetric.htm#chromaticQuasisymmetricMomentGraph","chromaticQuasisymmetricPowerSumExpansion":"chromaticQuasisymmetric.htm#chromaticQuasisymmetricPowerSumExpansion","chromaticQuasisymmetricProperties":"chromaticQuasisymmetric.htm#chromaticQuasisymmetricProperties","chromaticQuasisymmetricReciprocity":"chromaticQuasisymmetric.htm#chromaticQuasisymmetricReciprocity","chromaticQuasisymmetricSchurExpansion":"chromaticQuasisymmetric.htm#chromaticQuasisymmetricSchurExpansion","chromaticQuasisymmetricSymmetric":"chromaticQuasisymmetric.htm#chromaticQuasisymmetricSymmetric","chromaticQuasisymmetricUnitIntervalGraph":"chromaticQuasisymmetric.htm#chromaticQuasisymmetricUnitIntervalGraph","chromaticSymmetricFunction":"chromaticQuasisymmetric.htm#chromaticSymmetricFunction","chromaticWPolynomial":"chromaticQuasisymmetric.htm#chromaticWPolynomial","chromaticWordQuasisymmetricMatroid":"standardQuasiSymmetricFunctions.htm#chromaticWordQuasisymmetricMatroid","circularDyckDiagram":"chromaticQuasisymmetric.htm#circularDyckDiagram","circularUnitArcDigraph":"chromaticQuasisymmetric.htm#circularUnitArcDigraph","classicalMultisetEulerianRealRooted":"realRootedWords.htm#classicalMultisetEulerianRealRooted","classicalNarayanaPolynomialsRealRooted":"realRootedCatalan.htm#classicalNarayanaPolynomialsRealRooted","classicalRootCriteria":"realRooted.htm#classicalRootCriteria","classicalTypeEulerianRealRooted":"realRootedWords.htm#classicalTypeEulerianRealRooted","clawFreeGraph":"realRootedGraphs.htm#clawFreeGraph","clusterAlgebra":"algebras-and-rings.htm#clusterAlgebra","clusterAlgebraDefinition":"algebras-and-rings.htm#clusterAlgebraDefinition","cochargeStatistic":"kostkaFoulkes.htm#cochargeStatistic","coinversion":"macdonaldEperm.htm#coinversion","columnInsertion":"rsk.htm#columnInsertion","combinatorialObjects":"combinatorialObjects.htm#combinatorialObjects","combinatorialQuasisymmetricPowerSumBasis":"standardQuasiSymmetricFunctions.htm#combinatorialQuasisymmetricPowerSumBasis","combinatorialStablePolynomialExamples":"stablePolynomials.htm#combinatorialStablePolynomialExamples","commonInterlacing":"realRootedInterlacing.htm#commonInterlacing","commonInterlacingCompatibilityTheorem":"realRootedInterlacing.htm#commonInterlacingCompatibilityTheorem","companionMap":"preliminaries.htm#companionMap","compatiblePolynomialSet":"realRootedInterlacing.htm#compatiblePolynomialSet","compatiblePolynomials":"realRootedInterlacing.htm#compatiblePolynomials","completeFlagDefinition":"varieties.htm#completeFlagDefinition","completeFlaggedHomogeneous":"key.htm#completeFlaggedHomogeneous","completeFlaggedHomogeneousPolynomial":"key.htm#completeFlaggedHomogeneousPolynomial","completeH":"standardSymmetricFunctions.htm#completeH","completeHCylindric":"cylindricSchur.htm#completeHCylindric","completeHomogeneousSupersymmetricFunction":"superSymmetricSchur.htm#completeHomogeneousSupersymmetricFunction","completeMultipartite":"weightedChromatic.htm#completeMultipartite","compositionDefinition":"preliminaries.htm#compositionDefinition","compositionRefinementOrder":"standardQuasiSymmetricFunctions.htm#compositionRefinementOrder","compositionalHallLittlewoodPolynomial":"hallLittlewood.htm#compositionalHallLittlewoodPolynomial","compositionalShuffleConjecture":"diagonalHarmonics.htm#compositionalShuffleConjecture","compositionalShuffleTheorem":"diagonalHarmonics.htm#compositionalShuffleTheorem","compositions":"preliminaries.htm#compositions","computingKostkaFoulkes":"kostkaFoulkes.htm#computingKostkaFoulkes","computingSnCharacters":"representationTheory.htm#computingSnCharacters","configurationCharge":"riggedConfigurations.htm#configurationCharge","configurations":"riggedConfigurations.htm#configurations","conj:eulerianDelannoySquares":"realRooted.htm#conj:eulerianDelannoySquares","conjugatePartition":"preliminaries.htm#conjugatePartition","connectedMatroid":"matroids.htm#connectedMatroid","contentOfBox":"preliminaries.htm#contentOfBox","contingencyMatrices":"gtpatterns.htm#contingencyMatrices","coplacticCrystalOperator":"crystals.htm#coplacticCrystalOperator","coverRelation":"posets.htm#coverRelation","coxeterAffine":"coxeterGroups.htm#coxeterAffine","coxeterCombinatorics":"coxeterGroups.htm#coxeterCombinatorics","coxeterGroups":"coxeterGroups.htm#coxeterGroups","coxeterMatroid":"matroids.htm#coxeterMatroid","coxeterSystemDefinition":"coxeterGroups.htm#coxeterSystemDefinition","coxeterTypeA":"coxeterGroups.htm#coxeterTypeA","coxeterTypeB":"coxeterGroups.htm#coxeterTypeB","coxeterTypeD":"coxeterGroups.htm#coxeterTypeD","crossPolytope":"polytopes.htm#crossPolytope","crossPolytopeDefinition":"polytopes.htm#crossPolytopeDefinition","crossingCondition":"lgv-lemma.htm#crossingCondition","crystalDefinition":"crystals.htm#crystalDefinition","crystalGraphs":"crystals.htm#crystalGraphs","crystalHighestWeight":"crystals.htm#crystalHighestWeight","crystalIString":"crystals.htm#crystalIString","crystalLengthFunction":"crystals.htm#crystalLengthFunction","crystalLoweringOperator":"crystals.htm#crystalLoweringOperator","crystalOperators":"crystals.htm#crystalOperators","crystalRaisingOperator":"crystals.htm#crystalRaisingOperator","crystalReflection":"crystals.htm#crystalReflection","crystalReflectionOperator":"crystals.htm#crystalReflectionOperator","crystalSsyt":"crystals.htm#crystalSsyt","crystalWeightFunction":"crystals.htm#crystalWeightFunction","crystallographicCoxeterGroup":"coxeterGroups.htm#crystallographicCoxeterGroup","crystals":"crystals.htm#crystals","crystalsAndRSKI":"tableauOperators.htm#crystalsAndRSKI","crystalsAndRSKII":"tableauOperators.htm#crystalsAndRSKII","crystalsAndRSKIII":"tableauOperators.htm#crystalsAndRSKIII","crystalsRSK":"tableauOperators.htm#crystalsRSK","crystalsSeeAlso":"crystals.htm#crystalsSeeAlso","crystalsTypeB":"crystals.htm#crystalsTypeB","csp01Matrices":"cspMisc.htm#csp01Matrices","csp12":"cspCatalan.htm#csp12","csp12twist":"cspCatalan.htm#csp12twist","cspASM":"cspTableau.htm#cspASM","cspAndRepTheory":"cyclic-sieving.htm#cspAndRepTheory","cspAnnularNonCrossingPermutations":"cspMatch.htm#cspAnnularNonCrossingPermutations","cspAreaSequences":"cspPath.htm#cspAreaSequences","cspBinaryWordsTwist":"cspWord.htm#cspBinaryWordsTwist","cspCatalan":"cspCatalan.htm#cspCatalan","cspCatalanTriangulations":"cspCatalan.htm#cspCatalanTriangulations","cspCyclicCodes":"cspMisc.htm#cspCyclicCodes","cspDefinition":"cyclic-sieving.htm#cspDefinition","cspFansOfDyckPaths":"cspPath.htm#cspFansOfDyckPaths","cspGeneralSSYT":"cspTableau.htm#cspGeneralSSYT","cspGeneralizedClusterComplexes":"cspCatalan.htm#cspGeneralizedClusterComplexes","cspHookSSYT":"cspTableau.htm#cspHookSSYT","cspIncreasingTableau":"cspTableau.htm#cspIncreasingTableau","cspIndependentSetsGraphs":"cspMisc.htm#cspIndependentSetsGraphs","cspKrewerasWords":"cspWord.htm#cspKrewerasWords","cspLatticePolytopes":"cspMisc.htm#cspLatticePolytopes","cspLatticeWalks":"cspPath.htm#cspLatticeWalks","cspListFrieze":"cspMisc.htm#cspListFrieze","cspListMiscTableaux":"cspTableau.htm#cspListMiscTableaux","cspListPosetsMinuscule":"cspMisc.htm#cspListPosetsMinuscule","cspListPosetsRoot":"cspMisc.htm#cspListPosetsRoot","cspListSSTableaux":"cspTableau.htm#cspListSSTableaux","cspLongestTypeBReducedWords":"cspWord.htm#cspLongestTypeBReducedWords","cspLucasWords":"cspWord.htm#cspLucasWords","cspMacdonaldE":"cspTableau.htm#cspMacdonaldE","cspMatch":"cspMatch.htm#cspMatch","cspMatchings":"cspMatch.htm#cspMatchings","cspMaximalWeights":"cspMisc.htm#cspMaximalWeights","cspMisc":"cspMisc.htm#cspMisc","cspMultisets":"cspWord.htm#cspMultisets","cspNarayana":"cspCatalan.htm#cspNarayana","cspNonCrossingForests":"cspMatch.htm#cspNonCrossingForests","cspNonCrossingMatchingsI":"cspCatalan.htm#cspNonCrossingMatchingsI","cspNonCrossingPartitions":"cspCatalan.htm#cspNonCrossingPartitions","cspNonCrossingPartitionsFuss":"cspCatalan.htm#cspNonCrossingPartitionsFuss","cspPath":"cspPath.htm#cspPath","cspPathGraphLabelings":"cspWord.htm#cspPathGraphLabelings","cspPermutations":"cspWord.htm#cspPermutations","cspPermutations2":"cspWord.htm#cspPermutations2","cspPermutations3":"cspWord.htm#cspPermutations3","cspPlanePartitions":"cspTableau.htm#cspPlanePartitions","cspPlethysmCoefficients":"cspTableau.htm#cspPlethysmCoefficients","cspPolygon":"cspMatch.htm#cspPolygon","cspProper":"cyclic-sieving.htm#cspProper","cspRationalCatalan":"cspCatalan.htm#cspRationalCatalan","cspRectangularDominoTableaux":"cspTableau.htm#cspRectangularDominoTableaux","cspRectangularSSYT":"cspTableau.htm#cspRectangularSSYT","cspRectangularSYT":"cspTableau.htm#cspRectangularSYT","cspSignedRectangularSYT":"cspTableau.htm#cspSignedRectangularSYT","cspStaircaseSYT":"cspTableau.htm#cspStaircaseSYT","cspStretchedSYT":"cspTableau.htm#cspStretchedSYT","cspStrongDichotomy":"cspMisc.htm#cspStrongDichotomy","cspTableau":"cspTableau.htm#cspTableau","cspTensors":"cspMisc.htm#cspTensors","cspUniversal":"cyclic-sieving.htm#cspUniversal","cspWCatalan":"cspCatalan.htm#cspWCatalan","cspWord":"cspWord.htm#cspWord","cspWords":"cspWord.htm#cspWords","cspWordsTwist":"cspWord.htm#cspWordsTwist","cuiZhuWeightedPaths":"polyaFrequency.htm#cuiZhuWeightedPaths","cuttingStrip":"schur.htm#cuttingStrip","cyclage":"kostkaFoulkes.htm#cyclage","cyclageDefinition":"kostkaFoulkes.htm#cyclageDefinition","cycleIndexPolynomial":"cycleIndexPolynomial.htm#cycleIndexPolynomial","cycleIndexPolynomialDefinition":"cycleIndexPolynomial.htm#cycleIndexPolynomialDefinition","cyclicDescentMap":"q-analogs.htm#cyclicDescentMap","cyclicInductionSchurCone":"schurPositivity.htm#cyclicInductionSchurCone","cyclicIntervalDefinition":"positroids.htm#cyclicIntervalDefinition","cyclicPolytopeMagicPositivity":"ehrhart.htm#cyclicPolytopeMagicPositivity","cyclicQuasisymmetricFunction":"standardQuasiSymmetricFunctions.htm#cyclicQuasisymmetricFunction","cyclicSieving":"cyclic-sieving.htm#cyclicSieving","cyclotomicGeneratingFunction":"q-analogs.htm#cyclotomicGeneratingFunction","cylindricAffineLittlewood":"cylindricSchur.htm#cylindricAffineLittlewood","cylindricCompleteHomogeneousPolynomial":"cylindricSchur.htm#cylindricCompleteHomogeneousPolynomial","cylindricElementaryPolynomial":"cylindricSchur.htm#cylindricElementaryPolynomial","cylindricEquivariantQuantumPieri":"cylindricSchur.htm#cylindricEquivariantQuantumPieri","cylindricSchurDefinition":"cylindricSchur.htm#cylindricSchurDefinition","cylindricSchurFunction":"cylindricSchur.htm#cylindricSchurFunction","cylindricSchurPositivity":"cylindricSchur.htm#cylindricSchurPositivity","cylindricSchurSeeAlso":"cylindricSchur.htm#cylindricSchurSeeAlso","cylindricSemistandardTableau":"cylindricSchur.htm#cylindricSemistandardTableau","cylindricShape":"cylindricSchur.htm#cylindricShape","cylindricSkewShape":"cylindricSchur.htm#cylindricSkewShape","daggerAction":"unicellular-llt.htm#daggerAction","decreasingFactorization":"stanleySymmetric.htm#decreasingFactorization","decreasingPermutation":"stanleySymmetric.htm#decreasingPermutation","def:hookFilter":"symmetricFunctionsSpace.htm#def:hookFilter","def:koganFace":"key.htm#def:koganFace","def:reducedKogan":"key.htm#def:reducedKogan","delannoyRowPolynomials":"realRooted.htm#delannoyRowPolynomials","deltaMatroid":"matroids.htm#deltaMatroid","deltaOperator":"diagonalHarmonics.htm#deltaOperator","deltaPrimeOperator":"diagonalHarmonics.htm#deltaPrimeOperator","deltaProfile":"whittaker.htm#deltaProfile","deltaSpringerHallLittlewood":"hallLittlewood.htm#deltaSpringerHallLittlewood","deltaSpringerVariety":"hallLittlewood.htm#deltaSpringerVariety","demazureAtom":"key.htm#demazureAtom","demazureCharacter":"key.htm#demazureCharacter","demazureCrystal":"crystals.htm#demazureCrystal","demazureCrystals":"crystals.htm#demazureCrystals","demazureTAtom":"macdonaldEperm.htm#demazureTAtom","dependencePolynomial":"realRootedGraphs.htm#dependencePolynomial","derangedMap":"realRootedInterlacing.htm#derangedMap","derangementPart":"q-analogs.htm#derangementPart","derangementPolynomial":"realRootedWords.htm#derangementPolynomial","derangementPolynomialsRealRooted":"realRootedWords.htm#derangementPolynomialsRealRooted","derangementTransformation":"realRootedInterlacing.htm#derangementTransformation","derivativeRecurrenceRealRoots":"polynomialRecurrences.htm#derivativeRecurrenceRealRoots","derivedSchur":"schur.htm#derivedSchur","derivedSchurPolynomial":"schur.htm#derivedSchurPolynomial","descartesRuleOfSigns":"realRooted.htm#descartesRuleOfSigns","descentSet":"preliminaries.htm#descentSet","descentsMajorIndex":"preliminaries.htm#descentsMajorIndex","determinantalStability":"stablePolynomials.htm#determinantalStability","determinantalVarietyDefinition":"schurPositivity.htm#determinantalVarietyDefinition","deyInterlacingCriterion":"realRootedSetPartitions.htm#deyInterlacingCriterion","dhDefinitions":"diagonalHarmonics.htm#dhDefinitions","dhDelta":"shuffleAndDelta.htm#dhDelta","dhDeltaSquare":"shuffleAndDelta.htm#dhDeltaSquare","dhExtendedDelta":"shuffleAndDelta.htm#dhExtendedDelta","dhGeneralizedDelta":"shuffleAndDelta.htm#dhGeneralizedDelta","dhMonomial":"shuffleAndDelta.htm#dhMonomial","dhOperators":"diagonalHarmonics.htm#dhOperators","dhOther":"shuffleAndDelta.htm#dhOther","dhSchur":"shuffleAndDelta.htm#dhSchur","dhShuffle":"diagonalHarmonics.htm#dhShuffle","dhShuffleProps":"diagonalHarmonics.htm#dhShuffleProps","dhShuffleTheorem":"diagonalHarmonics.htm#dhShuffleTheorem","dhShuffleTheoremGen":"diagonalHarmonics.htm#dhShuffleTheoremGen","dhSquare":"shuffleAndDelta.htm#dhSquare","dhThetaNegut":"diagonalHarmonics.htm#dhThetaNegut","diSchur":"qsymSchur.htm#diSchur","diagonalCoinvariantExamplesSpecializations":"diagonalHarmonics.htm#diagonalCoinvariantExamplesSpecializations","diagonalCoinvariantHilbertFrobenius":"diagonalHarmonics.htm#diagonalCoinvariantHilbertFrobenius","diagonalCoinvariantRing":"diagonalHarmonics.htm#diagonalCoinvariantRing","diagonalHarmonics":"diagonalHarmonics.htm#diagonalHarmonics","diagonalHarmonicsSpace":"diagonalHarmonics.htm#diagonalHarmonicsSpace","diagonalSuperspaceCoinvariants":"diagonalHarmonics.htm#diagonalSuperspaceCoinvariants","diagonalSuperspaceSignComponent":"diagonalHarmonics.htm#diagonalSuperspaceSignComponent","diagramDefinition":"kohnert.htm#diagramDefinition","diagramWeight":"kohnert.htm#diagramWeight","diamondProduct":"realRooted.htm#diamondProduct","diamondProductDefinition":"realRooted.htm#diamondProductDefinition","dictionaryAlgebraRepresentation":"dictionary.htm#dictionaryAlgebraRepresentation","dictionaryCommutingVariables":"dictionary.htm#dictionaryCommutingVariables","dictionaryGeometryPolytopes":"dictionary.htm#dictionaryGeometryPolytopes","dictionaryPartitionsTableaux":"dictionary.htm#dictionaryPartitionsTableaux","dihedralSieving":"cyclic-sieving.htm#dihedralSieving","directSumPermutations":"permutationPatterns.htm#directSumPermutations","disjointUnionPoset":"realRooted.htm#disjointUnionPoset","distributiveLattice":"posets.htm#distributiveLattice","dividedDifferenceOperators":"schubert.htm#dividedDifferenceOperators","dominanceOrder":"preliminaries.htm#dominanceOrder","dominantSequenceOfRectangles":"kostkaFoulkes.htm#dominantSequenceOfRectangles","dominoInsertion":"rsk.htm#dominoInsertion","dotAction":"dot-action.htm#dotAction","dotActionDefinition":"dot-action.htm#dotActionDefinition","dotActionFrobeniusPipeline":"dot-action.htm#dotActionFrobeniusPipeline","dotActionGKMModel":"dot-action.htm#dotActionGKMModel","dotActionHessenbergSequence":"dot-action.htm#dotActionHessenbergSequence","dotActionOnGKMTuple":"dot-action.htm#dotActionOnGKMTuple","dotActionShareshianWachsTheorem":"dot-action.htm#dotActionShareshianWachsTheorem","dotActionTypeBC":"dot-action.htm#dotActionTypeBC","doubleGrothendieckPolynomial":"grothendieck.htm#doubleGrothendieckPolynomial","doubleHallLittlewoodPolynomial":"hallLittlewood.htm#doubleHallLittlewoodPolynomial","doubleJackPolynomial":"jack.htm#doubleJackPolynomial","doubleMacdonaldPolynomial":"macdonaldP.htm#doubleMacdonaldPolynomial","doubleStanleySymmetricFunction":"stanleySymmetric.htm#doubleStanleySymmetricFunction","doublySymmetricFunction":"superSymmetricSchur.htm#doublySymmetricFunction","dualEvacuation":"tableauOperators.htm#dualEvacuation","dualGrothendieckPercolation":"grothendieck.htm#dualGrothendieckPercolation","dualGrothendieckSchurExpansion":"grothendieck.htm#dualGrothendieckSchurExpansion","dualImmaculateDefinition":"qsymSchur.htm#dualImmaculateDefinition","dualImmaculateExamples":"qsymSchur.htm#dualImmaculateExamples","dualImmaculateHookFormula":"qsymSchur.htm#dualImmaculateHookFormula","dualImmaculateSchurFunction":"qsymSchur.htm#dualImmaculateSchurFunction","dualImmaculateVariants":"qsymSchur.htm#dualImmaculateVariants","dualInsertion":"rsk.htm#dualInsertion","dualKSchurFunction":"stanleySymmetric.htm#dualKSchurFunction","dualKnuthEquivalent":"rsk.htm#dualKnuthEquivalent","dualSSYT":"rsk.htm#dualSSYT","durfeeSize":"preliminaries.htm#durfeeSize","dyckPathStatisticPolynomialsRealRooted":"realRootedCatalan.htm#dyckPathStatisticPolynomialsRealRooted","dynkinReversal":"backStableSchubert.htm#dynkinReversal","edgePolytope":"polytopes.htm#edgePolytope","edgePolytopeDefinition":"polytopes.htm#edgePolytopeDefinition","edreiThomaTheorem":"polyaFrequency.htm#edreiThomaTheorem","ehrhart":"ehrhart.htm#ehrhart","ehrhartConjecturalPositivity":"ehrhart.htm#ehrhartConjecturalPositivity","ehrhartMacdonaldReciprocity":"polytopes.htm#ehrhartMacdonaldReciprocity","ehrhartOperations":"ehrhart.htm#ehrhartOperations","ehrhartPolynomialDefinition":"polytopes.htm#ehrhartPolynomialDefinition","ehrhartPolytopes":"polytopes.htm#ehrhartPolytopes","ehrhartPosets":"ehrhart.htm#ehrhartPosets","ehrhartQuasiPolynomial":"polytopes.htm#ehrhartQuasiPolynomial","ehrhartQuestions":"ehrhart.htm#ehrhartQuestions","ehrhartReciprocityHub":"ehrhart.htm#ehrhartReciprocityHub","ehrhartSeries":"polytopes.htm#ehrhartSeries","ehrhartTheory":"ehrhart.htm#ehrhartTheory","ehrhartWhereItAppears":"ehrhart.htm#ehrhartWhereItAppears","electronicMacdonaldPolynomial":"macdonaldP.htm#electronicMacdonaldPolynomial","elegantFilling":"grothendieck.htm#elegantFilling","elementaryDualKnuthTransformation":"rsk.htm#elementaryDualKnuthTransformation","elementaryE":"standardSymmetricFunctions.htm#elementaryE","elementaryECylindric":"cylindricSchur.htm#elementaryECylindric","elementaryKnuthTransform":"kostkaFoulkes.htm#elementaryKnuthTransform","elementarySupersymmetricFunction":"superSymmetricSchur.htm#elementarySupersymmetricFunction","elephantPolynomials":"realRootedInterlacing.htm#elephantPolynomials","ellipticSchur":"schur.htm#ellipticSchur","ellipticSchurFunction":"schur.htm#ellipticSchurFunction","enrichedPPartitions":"peakQuasisymmetric.htm#enrichedPPartitions","equivariantQuasisymmetry":"standardQuasiSymmetricFunctions.htm#equivariantQuasisymmetry","erdosSzekeresTheorem":"rsk.htm#erdosSzekeresTheorem","eulerian":"eulerian.htm#eulerian","eulerianGamma":"eulerian.htm#eulerianGamma","eulerianGeneralizations":"eulerian.htm#eulerianGeneralizations","eulerianGeneratingFunctions":"eulerian.htm#eulerianGeneratingFunctions","eulerianGeometry":"eulerian.htm#eulerianGeometry","eulerianPolynomial":"eulerian.htm#eulerianPolynomial","eulerianPolynomialDefinition":"eulerian.htm#eulerianPolynomialDefinition","eulerianRecurrences":"eulerian.htm#eulerianRecurrences","eulerianSymmetric":"eulerianSymmetric.htm#eulerianSymmetric","eulerianSymmetricCSP":"eulerianSymmetric.htm#eulerianSymmetricCSP","eulerianSymmetricColored":"eulerianSymmetric.htm#eulerianSymmetricColored","eulerianSymmetricDefinition":"eulerianSymmetric.htm#eulerianSymmetricDefinition","eulerianSymmetricPlethystic":"eulerianSymmetric.htm#eulerianSymmetricPlethystic","eulerianSymmetricPowerSumPositivity":"eulerianSymmetric.htm#eulerianSymmetricPowerSumPositivity","eulerianSymmetricSchurPositivity":"eulerianSymmetric.htm#eulerianSymmetricSchurPositivity","eulerianSymmetry":"eulerian.htm#eulerianSymmetry","eulerianTransformation":"realRooted.htm#eulerianTransformation","eulerianWorpitzky":"eulerian.htm#eulerianWorpitzky","eulerianZeros":"eulerian.htm#eulerianZeros","evacuation":"tableauOperators.htm#evacuation","evenOrthogonalSchurPolynomial":"schurSymplecticOrthogonal.htm#evenOrthogonalSchurPolynomial","evenPartition":"schur.htm#evenPartition","evenSymplecticSchurPolynomial":"schurSymplecticOrthogonal.htm#evenSymplecticSchurPolynomial","ex:eulerianSturm":"realRootedWords.htm#ex:eulerianSturm","ex:narayanaSturm":"realRootedCatalan.htm#ex:narayanaSturm","ex:peakSturm":"realRootedWords.htm#ex:peakSturm","ex:stableMultivariateEulerianOperator":"stablePolynomials.htm#ex:stableMultivariateEulerianOperator","ex:typeBEulerianSturm":"realRootedWords.htm#ex:typeBEulerianSturm","extSchur":"qsymSchur.htm#extSchur","extendedChromatic":"weightedChromatic.htm#extendedChromatic","extendedDeltaConjecture":"shuffleAndDelta.htm#extendedDeltaConjecture","extendedHikitaPolynomial":"shuffleAndDelta.htm#extendedHikitaPolynomial","extendedRationalShuffleConjecture":"shuffleAndDelta.htm#extendedRationalShuffleConjecture","extendedSchurFunction":"qsymSchur.htm#extendedSchurFunction","externallyActive":"matroids.htm#externallyActive","factorialHookSchur":"superSymmetricSchur.htm#factorialHookSchur","factorialLLTPolynomial":"llt.htm#factorialLLTPolynomial","factorialSchurFunctionDefinition":"schurShifted.htm#factorialSchurFunctionDefinition","fallingFactorialDefinition":"schurShifted.htm#fallingFactorialDefinition","favardInterlacing":"realRootedInterlacing.htm#favardInterlacing","ferrariMartinPairingAlgorithm":"macdonaldE.htm#ferrariMartinPairingAlgorithm","fflvPolytope":"gtpatterns.htm#fflvPolytope","findstatPermutationCSP":"cyclic-sieving.htm#findstatPermutationCSP","finiteMultiplierSequenceCriterion":"realRooted.htm#finiteMultiplierSequenceCriterion","finitePolyaFrequencySequence":"polyaFrequency.htm#finitePolyaFrequencySequence","firstDerivativeCriterion":"polynomialRecurrences.htm#firstDerivativeCriterion","fixedPointFreeStanleySymmetricFunction":"stanleySymmetric.htm#fixedPointFreeStanleySymmetricFunction","flagMatroid":"matroids.htm#flagMatroid","flagVarieties":"varieties.htm#flagVarieties","flagVarietyDefinition":"varieties.htm#flagVarietyDefinition","flaggedFactorialGrothendieckPolynomial":"grothendieck.htm#flaggedFactorialGrothendieckPolynomial","flaggedKostkaCoefficientDefinition":"schurFlagged.htm#flaggedKostkaCoefficientDefinition","flaggedLLT":"llt.htm#flaggedLLT","flaggedLLTPolynomialDefinition":"llt.htm#flaggedLLTPolynomialDefinition","flaggedPPartition":"pPartitions.htm#flaggedPPartition","flaggedRefinedSkewStableGrothendieck":"grothendieck.htm#flaggedRefinedSkewStableGrothendieck","flaggedSchurDividedDifferences":"schurFlagged.htm#flaggedSchurDividedDifferences","flaggedSchurKeyExpansion":"schurFlagged.htm#flaggedSchurKeyExpansion","flaggedSchurKostka":"schurFlagged.htm#flaggedSchurKostka","flaggedSchurPolynomialDefinition":"schurFlagged.htm#flaggedSchurPolynomialDefinition","flaggedSchurPolynomialJacobiTrudi":"schurFlagged.htm#flaggedSchurPolynomialJacobiTrudi","flaggedSchurRootsOfUnity":"schurFlagged.htm#flaggedSchurRootsOfUnity","flaggedSetValuedTableau":"grothendieck.htm#flaggedSetValuedTableau","flaggedSkewSchurPolynomialDefinition":"schurFlagged.htm#flaggedSkewSchurPolynomialDefinition","flaggedWeylModule":"schubert.htm#flaggedWeylModule","flatComposition":"assaf.htm#flatComposition","flippedExtSchur":"qsymSchur.htm#flippedExtSchur","flippedExtendedSchurFunction":"qsymSchur.htm#flippedExtendedSchurFunction","flowPolytope":"polytopes.htm#flowPolytope","flowPolytopeDefinition":"polytopes.htm#flowPolytopeDefinition","foataStrehlAction":"gammaPositivity.htm#foataStrehlAction","fockSpace":"symmetricFunctionsSpace.htm#fockSpace","fominKirillovConjecture":"schubert.htm#fominKirillovConjecture","forestPolynomialDefinition":"assaf.htm#forestPolynomialDefinition","forestPolynomials":"assaf.htm#forestPolynomials","forgotten":"standardSymmetricFunctions.htm#forgotten","foulkesConjecture":"plethysm.htm#foulkesConjecture","frobeniusCharacteristic":"representationTheory.htm#frobeniusCharacteristic","frobeniusCoordinates":"preliminaries.htm#frobeniusCoordinates","frobeniusImage":"representationTheory.htm#frobeniusImage","fullyInterlacingLace":"realRootedInterlacing.htm#fullyInterlacingLace","fullyInterlacingSequence":"realRootedInterlacing.htm#fullyInterlacingSequence","fultonsConjecture":"littlewoodRichardson.htm#fultonsConjecture","fundamentalParticle":"assaf.htm#fundamentalParticle","fundamentalParticleBasis":"assaf.htm#fundamentalParticleBasis","fundamentalSlidePolynomialDefinition":"assaf.htm#fundamentalSlidePolynomialDefinition","fusionCoefficient":"cylindricSchur.htm#fusionCoefficient","fussCatalanObjects":"combinatorialObjects.htm#fussCatalanObjects","galConjecture":"gammaPositivity.htm#galConjecture","gammaPolynomial":"gammaPositivity.htm#gammaPolynomial","gammaPositiveExamples":"gammaPositivity.htm#gammaPositiveExamples","gammaPositivePolynomial":"gammaPositivity.htm#gammaPositivePolynomial","gammaPositivity":"gammaPositivity.htm#gammaPositivity","gammaPositivityMethods":"gammaPositivity.htm#gammaPositivityMethods","gardingPolynomial":"stablePolynomials.htm#gardingPolynomial","gardingPolynomials":"stablePolynomials.htm#gardingPolynomials","generalLinearGroup":"representationTheory.htm#generalLinearGroup","generalUsefulThings":"resources.htm#generalUsefulThings","generalizedDeltaSquareConjecture":"shuffleAndDelta.htm#generalizedDeltaSquareConjecture","generalizedDemazureAtoms":"generalizedDemazureAtoms.htm#generalizedDemazureAtoms","generalizedDualStableGrothendieckPolynomial":"grothendieck.htm#generalizedDualStableGrothendieckPolynomial","generalizedFactorialSchurFunction":"schurShifted.htm#generalizedFactorialSchurFunction","generalizedFlaggedSchurModule":"representationTheory.htm#generalizedFlaggedSchurModule","generalizedNarayanaPolynomialsRealRooted":"realRootedCatalan.htm#generalizedNarayanaPolynomialsRealRooted","generalizedNets":"chromaticQuasisymmetric.htm#generalizedNets","generalizedPermutohedron":"polytopes.htm#generalizedPermutohedron","generalizedPermutohedronDefinition":"polytopes.htm#generalizedPermutohedronDefinition","generalizedPitmanStanleyPolytope":"polytopes.htm#generalizedPitmanStanleyPolytope","generalizedSchurP":"schurPQ.htm#generalizedSchurP","generalizedSchurPFunction":"schurPQ.htm#generalizedSchurPFunction","generalizedSchurQ":"schurPQ.htm#generalizedSchurQ","generalizedSchurQFunction":"schurPQ.htm#generalizedSchurQFunction","generalizedSnakePoset":"realRooted.htm#generalizedSnakePoset","generalizedSnakeRealRooted":"realRooted.htm#generalizedSnakeRealRooted","generalizedSturmSequence":"realRootedInterlacing.htm#generalizedSturmSequence","genericPipeDream":"schubert.htm#genericPipeDream","geometricCrystal":"whittaker.htm#geometricCrystal","geometricRSK":"rsk.htm#geometricRSK","geometricRSKCorrespondence":"rsk.htm#geometricRSKCorrespondence","geometricWhittaker":"whittaker.htm#geometricWhittaker","gessel":"gessel.htm#gessel","gesselDefinitionCompositions":"gessel.htm#gesselDefinitionCompositions","gesselDefinitionSubsets":"gessel.htm#gesselDefinitionSubsets","gesselLengthSpecialization":"gessel.htm#gesselLengthSpecialization","gesselPowerSum":"gessel.htm#gesselPowerSum","gesselProduct":"gessel.htm#gesselProduct","gesselProperties":"gessel.htm#gesselProperties","gesselQuasiPowerSum":"gessel.htm#gesselQuasiPowerSum","gesselSlinkyRule":"gessel.htm#gesselSlinkyRule","gesselSpecializations":"gessel.htm#gesselSpecializations","gesselZeroHeckeAlgebra":"gessel.htm#gesselZeroHeckeAlgebra","gkmDescription":"dot-action.htm#gkmDescription","glCharacter":"representationTheory.htm#glCharacter","glKashiwaraCrystal":"crystals.htm#glKashiwaraCrystal","glRepresentation":"representationTheory.htm#glRepresentation","glidePolynomial":"assaf.htm#glidePolynomial","gorenstein":"polytopes.htm#gorenstein","gorensteinPolytope":"polytopes.htm#gorensteinPolytope","gradedPoset":"posets.htm#gradedPoset","graphAssociahedron":"polytopes.htm#graphAssociahedron","graphicMatroid":"matroids.htm#graphicMatroid","grassmannGrothendieckPolynomialDefinition":"grothendieck.htm#grassmannGrothendieckPolynomialDefinition","grassmannNecklaceDefinition":"positroids.htm#grassmannNecklaceDefinition","grassmannian":"schubert-calculus.htm#grassmannian","grassmannianDefinition":"schubert-calculus.htm#grassmannianDefinition","grassmannianPermutation":"schubert.htm#grassmannianPermutation","grassmannianSchubertClassDefinition":"schubert-calculus.htm#grassmannianSchubertClassDefinition","grassmannianSchubertClasses":"schubert-calculus.htm#grassmannianSchubertClasses","grassmannianSchubertDictionary":"schubert-calculus.htm#grassmannianSchubertDictionary","grassmannianVarieties":"varieties.htm#grassmannianVarieties","grassmannianVarietyDefinition":"varieties.htm#grassmannianVarietyDefinition","gromovWittenInvariant":"cylindricSchur.htm#gromovWittenInvariant","grothendieck":"grothendieck.htm#grothendieck","grothendieckASM":"grothendieck.htm#grothendieckASM","grothendieckCanonical":"grothendieck.htm#grothendieckCanonical","grothendieckDouble":"grothendieck.htm#grothendieckDouble","grothendieckDualStable":"grothendieck.htm#grothendieckDualStable","grothendieckDualStableBialternant":"grothendieck.htm#grothendieckDualStableBialternant","grothendieckDualStableJacobiTrudi":"grothendieck.htm#grothendieckDualStableJacobiTrudi","grothendieckDualStableLittlewood":"grothendieck.htm#grothendieckDualStableLittlewood","grothendieckFactorial":"grothendieck.htm#grothendieckFactorial","grothendieckGrassmann":"grothendieck.htm#grothendieckGrassmann","grothendieckGroup":"k-theory.htm#grothendieckGroup","grothendieckOperatorformula":"grothendieck.htm#grothendieckOperatorformula","grothendieckOrthodontia":"grothendieck.htm#grothendieckOrthodontia","grothendieckPipedreams":"grothendieck.htm#grothendieckPipedreams","grothendieckPolynomialDefinition":"grothendieck.htm#grothendieckPolynomialDefinition","grothendieckQuantumDouble":"grothendieck.htm#grothendieckQuantumDouble","grothendieckRefinedDualStable":"grothendieck.htm#grothendieckRefinedDualStable","grothendieckSkewStable":"grothendieck.htm#grothendieckSkewStable","grothendieckStable":"grothendieck.htm#grothendieckStable","grothendieckStableJacobiTrudi":"grothendieck.htm#grothendieckStableJacobiTrudi","grothendieckStableLittlewoodRichardson":"grothendieck.htm#grothendieckStableLittlewoodRichardson","grothendieckStableMN":"grothendieck.htm#grothendieckStableMN","grothendieckStableMoreInfo":"grothendieck.htm#grothendieckStableMoreInfo","grothendieckStablePieri":"grothendieck.htm#grothendieckStablePieri","grothendieckStableSchurExpansion":"grothendieck.htm#grothendieckStableSchurExpansion","grothendieckStableWeyl":"grothendieck.htm#grothendieckStableWeyl","grothendieckStableZeroHecke":"grothendieck.htm#grothendieckStableZeroHecke","grothendieckTableauxGeometry":"grothendieck.htm#grothendieckTableauxGeometry","groundSet":"matroids.htm#groundSet","groupRepresentation":"representationTheory.htm#groupRepresentation","gtApplications":"gtpatterns.htm#gtApplications","gtEhrhart":"gtpatterns.htm#gtEhrhart","gtHL":"gtpatterns.htm#gtHL","gtKey":"gtpatterns.htm#gtKey","gtKostka":"gtpatterns.htm#gtKostka","gtKostkaEhrhart":"gtpatterns.htm#gtKostkaEhrhart","gtKostkaInterior":"gtpatterns.htm#gtKostkaInterior","gtKostkaSaturation":"gtpatterns.htm#gtKostkaSaturation","gtKostkaVertices":"gtpatterns.htm#gtKostkaVertices","gtLR":"gtpatterns.htm#gtLR","gtLRInequalities":"gtpatterns.htm#gtLRInequalities","gtPattern":"gtpatterns.htm#gtPattern","gtPatternWeight":"gtpatterns.htm#gtPatternWeight","gtPolytope":"gtpatterns.htm#gtPolytope","gtPoset":"gtpatterns.htm#gtPoset","gtSchubert":"gtpatterns.htm#gtSchubert","gtSchur":"gtpatterns.htm#gtSchur","gtSemigroupRing":"gtpatterns.htm#gtSemigroupRing","gtpatterns":"gtpatterns.htm#gtpatterns","gtpatternsAsSSYT":"gtpatterns.htm#gtpatternsAsSSYT","gtpatternsTokuyama":"gtpatterns.htm#gtpatternsTokuyama","gtpolytopes":"gtpatterns.htm#gtpolytopes","gtpolytopesFVectors":"gtpatterns.htm#gtpolytopesFVectors","gtpolytopesMarkedPoset":"gtpatterns.htm#gtpolytopesMarkedPoset","gtpolytopesTypeBCD":"gtpatterns.htm#gtpolytopesTypeBCD","gurvitsCapacity":"stablePolynomials.htm#gurvitsCapacity","hFlaggedSchurDefinition":"schurFlagged.htm#hFlaggedSchurDefinition","hFlaggedSchurPolynomials":"schurFlagged.htm#hFlaggedSchurPolynomials","hStarInequalities":"ehrhart.htm#hStarInequalities","hStarPolynomial":"polytopes.htm#hStarPolynomial","hStarVector":"polytopes.htm#hStarVector","hadamardProduct":"realRooted.htm#hadamardProduct","hadamardProductDefinition":"realRooted.htm#hadamardProductDefinition","hadamardProductTheorems":"realRooted.htm#hadamardProductTheorems","haglundMacdonaldConjecture":"macdonaldP.htm#haglundMacdonaldConjecture","halfInterlacingEulerian":"realRootedWords.htm#halfInterlacingEulerian","hallInnerProduct":"symmetricFunctionsSpace.htm#hallInnerProduct","hallLittleWoodPWeylformula":"hallLittlewood.htm#hallLittleWoodPWeylformula","hallLittlewoodBhattacharya":"hallLittlewood.htm#hallLittlewoodBhattacharya","hallLittlewoodCauchy":"hallLittlewood.htm#hallLittlewoodCauchy","hallLittlewoodConjectures":"hallLittlewood.htm#hallLittlewoodConjectures","hallLittlewoodGT":"hallLittlewood.htm#hallLittlewoodGT","hallLittlewoodH":"hallLittlewood.htm#hallLittlewoodH","hallLittlewoodInnerProduct":"hallLittlewood.htm#hallLittlewoodInnerProduct","hallLittlewoodNegativeT":"hallLittlewood.htm#hallLittlewoodNegativeT","hallLittlewoodOtherCoeffs":"hallLittlewood.htm#hallLittlewoodOtherCoeffs","hallLittlewoodOtherRootSystems":"hallLittlewood.htm#hallLittlewoodOtherRootSystems","hallLittlewoodP":"hallLittlewood.htm#hallLittlewoodP","hallLittlewoodPieri":"hallLittlewood.htm#hallLittlewoodPieri","hallLittlewoodProducts":"hallLittlewood.htm#hallLittlewoodProducts","hallLittlewoodQ":"hallLittlewood.htm#hallLittlewoodQ","hallLittlewoodQPolynomial":"hallLittlewood.htm#hallLittlewoodQPolynomial","hallLittlewoodS":"hallLittlewood.htm#hallLittlewoodS","hallLittlewoodSFunction":"hallLittlewood.htm#hallLittlewoodSFunction","hallLittlewoodSFunctions":"hallLittlewood.htm#hallLittlewoodSFunctions","hallLittlewoodSchubertSeries":"hallLittlewood.htm#hallLittlewoodSchubertSeries","hallLittlewoodSchubertSeriesDefinition":"hallLittlewood.htm#hallLittlewoodSchubertSeriesDefinition","hallLittlewoodT":"hallLittlewood.htm#hallLittlewoodT","hallPolynomial":"hallLittlewood.htm#hallPolynomial","hallPolynomials":"hallLittlewood.htm#hallPolynomials","hankelTNN":"totalPositivity.htm#hankelTNN","hankelTotalPositivityContinuedFractions":"totalPositivity.htm#hankelTotalPositivityContinuedFractions","harrisWillenbring":"littlewoodRichardson.htm#harrisWillenbring","hasseDiagram":"posets.htm#hasseDiagram","heckeAlgebra":"algebras-and-rings.htm#heckeAlgebra","hermiteBiehlerTheorem":"realRootedInterlacing.htm#hermiteBiehlerTheorem","hermiteMatrix":"realRooted.htm#hermiteMatrix","hermitePoulainInterlacing":"realRootedInterlacing.htm#hermitePoulainInterlacing","hermitePoulainTheorem":"realRootedInterlacing.htm#hermitePoulainTheorem","hermiteSylvesterTheorem":"realRooted.htm#hermiteSylvesterTheorem","hessenbergFunction":"dot-action.htm#hessenbergFunction","hessenbergFunctionDefinition":"varieties.htm#hessenbergFunctionDefinition","hessenbergSequence":"dot-action.htm#hessenbergSequence","hessenbergVarieties":"dot-action.htm#hessenbergVarieties","hessenbergVarietiesBackground":"varieties.htm#hessenbergVarietiesBackground","hessenbergVarietyDefinition":"varieties.htm#hessenbergVarietyDefinition","hhlFillingFormula":"macdonaldE.htm#hhlFillingFormula","hibiLiFaceNumberConjecture":"polytopes.htm#hibiLiFaceNumberConjecture","higherLieCharacterDefinition":"cycleIndexPolynomial.htm#higherLieCharacterDefinition","higherSpechtPolynomials":"specht-modules.htm#higherSpechtPolynomials","highestReachableVertex":"unicellular-llt.htm#highestReachableVertex","hikita":"shuffleAndDelta.htm#hikita","hikitaGeometrySpecializations":"shuffleAndDelta.htm#hikitaGeometrySpecializations","hikitaOperators":"shuffleAndDelta.htm#hikitaOperators","hikitaPolynomial":"shuffleAndDelta.htm#hikitaPolynomial","hikitaPolynomialDefinition":"shuffleAndDelta.htm#hikitaPolynomialDefinition","hilbertSeries":"representationTheory.htm#hilbertSeries","hillmanGrassl":"rsk.htm#hillmanGrassl","hillmanGrasslCorrespondence":"rsk.htm#hillmanGrasslCorrespondence","hitPolynomial":"realRootedGraphs.htm#hitPolynomial","hiveDefinition":"hivePolytopes.htm#hiveDefinition","hivePolytope":"hivePolytopes.htm#hivePolytope","hivePolytopes":"hivePolytopes.htm#hivePolytopes","hiveRelatedModels":"hivePolytopes.htm#hiveRelatedModels","hiveSaturation":"hivePolytopes.htm#hiveSaturation","hiveStretching":"hivePolytopes.htm#hiveStretching","hivert":"hivert.htm#hivert","holmesCharacterRecursion":"murnaghanNakayama.htm#holmesCharacterRecursion","holmesChi":"murnaghanNakayama.htm#holmesChi","homogeneousRepresentation":"representationTheory.htm#homogeneousRepresentation","homomesy":"cyclic-sieving.htm#homomesy","hookFormula":"preliminaries.htm#hookFormula","hookLength":"preliminaries.htm#hookLength","hookSchur":"superSymmetricSchur.htm#hookSchur","hookSchurFunction":"superSymmetricSchur.htm#hookSchurFunction","hookSchurPlethDef":"superSymmetricSchur.htm#hookSchurPlethDef","hookSchurQS":"superSymmetricSchur.htm#hookSchurQS","hookSchurTableauDef":"superSymmetricSchur.htm#hookSchurTableauDef","hopfAlgebra":"algebras-and-rings.htm#hopfAlgebra","hopfAlgebraDefinition":"algebras-and-rings.htm#hopfAlgebraDefinition","hopfAlgebraSymmetricFunctions":"symmetricFunctionsSpace.htm#hopfAlgebraSymmetricFunctions","hopfChain":"shuffleAndDelta.htm#hopfChain","hornInequalities":"littlewoodRichardson.htm#hornInequalities","hourglassPlabicGraph":"cspTableau.htm#hourglassPlabicGraph","howToCite":"about.htm#howToCite","hurwitzImpliesCoefficientPositivity":"realRootedInterlacing.htm#hurwitzImpliesCoefficientPositivity","hurwitzLaceCriterion":"realRootedInterlacing.htm#hurwitzLaceCriterion","hurwitzMatrix":"realRootedInterlacing.htm#hurwitzMatrix","hypersimplex":"polytopes.htm#hypersimplex","hypersimplexDefinition":"polytopes.htm#hypersimplexDefinition","imathHallLittlewoodFunction":"hallLittlewood.htm#imathHallLittlewoodFunction","imathHallLittlewoodH":"hallLittlewood.htm#imathHallLittlewoodH","immaculateHook":"qsymSchur.htm#immaculateHook","immaculateSchur":"nonCommutativeFunctions.htm#immaculateSchur","immaculateSchurFunctionDefinition":"nonCommutativeFunctions.htm#immaculateSchurFunctionDefinition","immaculateTableau":"qsymSchur.htm#immaculateTableau","indCharBasis":"characterSymmetricFunctions.htm#indCharBasis","indTrivBasis":"characterSymmetricFunctions.htm#indTrivBasis","independencePolynomial":"realRootedGraphs.htm#independencePolynomial","infiniteQuasisymmetricGrassmannian":"standardQuasiSymmetricFunctions.htm#infiniteQuasisymmetricGrassmannian","infinitelyLogConcave":"realRooted.htm#infinitelyLogConcave","innerTensorProduct":"schur.htm#innerTensorProduct","insertionTableau":"rsk.htm#insertionTableau","integerDecompositionProperty":"polytopes.htm#integerDecompositionProperty","integralCyclicPolytope":"ehrhart.htm#integralCyclicPolytope","integralFormJackPolynomial":"jack.htm#integralFormJackPolynomial","integralFormMacdonaldPolynomial":"macdonaldP.htm#integralFormMacdonaldPolynomial","integralPolymatroidBase":"matroids.htm#integralPolymatroidBase","integralPolytope":"polytopes.htm#integralPolytope","interlacesPolynomial":"realRootedInterlacing.htm#interlacesPolynomial","interlacingFamilies":"realRootedInterlacing.htm#interlacingFamilies","interlacingPolynomials":"realRootedInterlacing.htm#interlacingPolynomials","interlacingSequenceDefinition":"realRootedInterlacing.htm#interlacingSequenceDefinition","interlacingSequences":"realRootedInterlacing.htm#interlacingSequences","interlacingTransforms":"realRootedInterlacing.htm#interlacingTransforms","interleavesPolynomial":"realRootedInterlacing.htm#interleavesPolynomial","intermediateMacdonaldPolynomial":"macdonaldP.htm#intermediateMacdonaldPolynomial","internallyActive":"matroids.htm#internallyActive","intersections":"various-research.htm#intersections","intervalOrder":"posets.htm#intervalOrder","invariantFlagGeneratingFunction":"whittaker.htm#invariantFlagGeneratingFunction","inverseKostkaMatrix":"standardSymmetricFunctions.htm#inverseKostkaMatrix","inversionNumber":"preliminaries.htm#inversionNumber","inversionSequence":"realRootedWords.htm#inversionSequence","inversionSequenceAscent":"realRootedWords.htm#inversionSequenceAscent","inversionSequenceCollision":"realRootedWords.htm#inversionSequenceCollision","inversionSequenceDescent":"realRootedWords.htm#inversionSequenceDescent","inversionSequenceEulerianRealRooted":"realRootedWords.htm#inversionSequenceEulerianRealRooted","inversionSet":"preliminaries.htm#inversionSet","inversionTableDefinition":"permutations.htm#inversionTableDefinition","inversionTriple":"macdonaldEperm.htm#inversionTriple","involutionPipeDream":"schubertVariations.htm#involutionPipeDream","involutionStanleySymmetricFunction":"stanleySymmetric.htm#involutionStanleySymmetricFunction","irreducibleRepresentationDefinition":"representationTheory.htm#irreducibleRepresentationDefinition","isobaricDividedDifferenceOperator":"key.htm#isobaricDividedDifferenceOperator","jackAffineAnalogues":"jack.htm#jackAffineAnalogues","jackCharacter":"jack.htm#jackCharacter","jackCharacterConjecture":"jack.htm#jackCharacterConjecture","jackCharacters":"jack.htm#jackCharacters","jackChromaticPowerSumExpansion":"jack.htm#jackChromaticPowerSumExpansion","jackCombinatorialFormula":"jack.htm#jackCombinatorialFormula","jackDerangement":"jack.htm#jackDerangement","jackDouble":"jack.htm#jackDouble","jackGesselExpansionConjecture":"jack.htm#jackGesselExpansionConjecture","jackHitPolynomial":"jack.htm#jackHitPolynomial","jackInnerProduct":"jack.htm#jackInnerProduct","jackJ":"jack.htm#jackJ","jackJCauchy":"jack.htm#jackJCauchy","jackJiMonomialPositivity":"jackF.htm#jackJiMonomialPositivity","jackLaplaceBeltrami":"jack.htm#jackLaplaceBeltrami","jackLittlewoodRichardsonCoefficient":"jack.htm#jackLittlewoodRichardsonCoefficient","jackLittlewoodRichardsonConjecture":"jack.htm#jackLittlewoodRichardsonConjecture","jackP":"jack.htm#jackP","jackPCauchy":"jack.htm#jackPCauchy","jackPi":"jackF.htm#jackPi","jackPiDef":"jackF.htm#jackPiDef","jackPiLR":"jackF.htm#jackPiLR","jackPieri":"jack.htm#jackPieri","jackRecursive":"jack.htm#jackRecursive","jackRookPolynomial":"jack.htm#jackRookPolynomial","jackSchurExpansionConjecture":"jack.htm#jackSchurExpansionConjecture","jackShifted":"jackShifted.htm#jackShifted","jackShiftedDef":"jackShifted.htm#jackShiftedDef","jackShiftedLittlewoodRichardson":"jackShifted.htm#jackShiftedLittlewoodRichardson","jackShiftedLittlewoodRichardsonConjecture":"jackShifted.htm#jackShiftedLittlewoodRichardsonConjecture","jackShiftedLittlewoodRichardsonData":"jackShifted.htm#jackShiftedLittlewoodRichardsonData","jackShiftedLittlewoodRichardsonFormula":"jackShifted.htm#jackShiftedLittlewoodRichardsonFormula","jackShiftedRSSYTDef":"jackShifted.htm#jackShiftedRSSYTDef","jackSkew":"jack.htm#jackSkew","jackSpecializations":"jack.htm#jackSpecializations","jackTableauFormula":"jack.htm#jackTableauFormula","jacobiStirlingDescentRealRooted":"realRootedSetPartitions.htm#jacobiStirlingDescentRealRooted","jacobiStirlingSpecializations":"standardSymmetricFunctions.htm#jacobiStirlingSpecializations","jamesKerberBranchingRule":"murnaghanNakayama.htm#jamesKerberBranchingRule","jeuDeTaquin":"tableauOperators.htm#jeuDeTaquin","jeuDeTaquinOperation":"tableauOperators.htm#jeuDeTaquinOperation","jochemkoVeroneseSymmetricDecomposition":"realRootedInterlacing.htm#jochemkoVeroneseSymmetricDecomposition","jordanHolderSet":"posets.htm#jordanHolderSet","kBoundedPartition":"kschur.htm#kBoundedPartition","kKSchurPositivity":"kschur.htm#kKSchurPositivity","kKohnertDiagram":"lascoux.htm#kKohnertDiagram","kPromotionOperator":"tableauOperators.htm#kPromotionOperator","kRibbonTableaux":"borderStripTableaux.htm#kRibbonTableaux","kSchur":"kschur.htm#kSchur","kSchurDefinition":"kschur.htm#kSchurDefinition","kSchurDuality":"kschur.htm#kSchurDuality","kSchurFunctionDefinition":"kschur.htm#kSchurFunctionDefinition","kSchurLittlewoodRichardson":"kschur.htm#kSchurLittlewoodRichardson","kSchurMacdonald":"kschur.htm#kSchurMacdonald","kSchurMurnaghanNakayama":"kschur.htm#kSchurMurnaghanNakayama","kSchurPieri":"kschur.htm#kSchurPieri","kSchurSchurExpansion":"kschur.htm#kSchurSchurExpansion","kTheoreticSchubertCalculus":"k-theory.htm#kTheoreticSchubertCalculus","kTheory":"k-theory.htm#kTheory","kTheoryCoherentSheaves":"k-theory.htm#kTheoryCoherentSheaves","kTheoryHowToRead":"k-theory.htm#kTheoryHowToRead","kaonBasis":"lascoux.htm#kaonBasis","kashiwaraCrystals":"crystals.htm#kashiwaraCrystals","kashiwaraNakashimaTableau":"schurSymplecticOrthogonal.htm#kashiwaraNakashimaTableau","katalanSymmetric":"kschur.htm#katalanSymmetric","katalanSymmetricFunction":"kschur.htm#katalanSymmetricFunction","kazhdanLusztigPolynomials":"algebras-and-rings.htm#kazhdanLusztigPolynomials","key":"key.htm#key","keyDefinitionGT":"key.htm#keyDefinitionGT","keyDefinitionKeyFillings":"key.htm#keyDefinitionKeyFillings","keyDefinitionOperators":"key.htm#keyDefinitionOperators","keyDefinitionSSYT":"key.htm#keyDefinitionSSYT","keyDiagram":"kohnert.htm#keyDiagram","keyEhrhart":"key.htm#keyEhrhart","keyFromKostant":"key.htm#keyFromKostant","keyGelfandTsetlinFormula":"key.htm#keyGelfandTsetlinFormula","keyKempf":"key.htm#keyKempf","keyKogan":"key.htm#keyKogan","keyModule":"representationTheory.htm#keyModule","keyPieri":"key.htm#keyPieri","keyPolynomial":"key.htm#keyPolynomial","keyPolynomialOperatorDefinition":"key.htm#keyPolynomialOperatorDefinition","keyProduct":"key.htm#keyProduct","keyRealRootedHstar":"various-research.htm#keyRealRootedHstar","keyRepresentation":"key.htm#keyRepresentation","keyRightKeyDecreasingWords":"key.htm#keyRightKeyDecreasingWords","keySkew":"key.htm#keySkew","keyTableau":"key.htm#keyTableau","keyTableauSSYTDefinition":"key.htm#keyTableauSSYTDefinition","keyTypeBC":"key.htm#keyTypeBC","keysAndAtomsTypeBCD":"key.htm#keysAndAtomsTypeBCD","klyachkoAlgebra":"algebras-and-rings.htm#klyachkoAlgebra","knopSahiRecurrence":"macdonaldE.htm#knopSahiRecurrence","knuthEquivalent":"kostkaFoulkes.htm#knuthEquivalent","knuthRelations":"rsk.htm#knuthRelations","knuthRelationsDual":"rsk.htm#knuthRelationsDual","knutsonTaoHoneycomb":"littlewoodRichardson.htm#knutsonTaoHoneycomb","knutsonTaoWoodwardPuzzles":"littlewoodRichardson.htm#knutsonTaoWoodwardPuzzles","koganFace":"key.htm#koganFace","koganType":"key.htm#koganType","kohnert":"kohnert.htm#kohnert","kohnertDefinition":"kohnert.htm#kohnertDefinition","kohnertMove":"kohnert.htm#kohnertMove","kohnertPolynomial":"kohnert.htm#kohnertPolynomial","kohnertPolynomialDefinition":"kohnert.htm#kohnertPolynomialDefinition","kohnertQuasi":"kohnert.htm#kohnertQuasi","kohnertQuasiKey":"kohnert.htm#kohnertQuasiKey","kohnertQuasisymmetricFunction":"kohnert.htm#kohnertQuasisymmetricFunction","kohnertTableauDefinition":"kohnert.htm#kohnertTableauDefinition","kohnertTableaux":"kohnert.htm#kohnertTableaux","koikeTeradaBasis":"littlewoodRichardson.htm#koikeTeradaBasis","koornwinderCombinatorialFormulas":"koornwinder.htm#koornwinderCombinatorialFormulas","koornwinderDefinition":"koornwinder.htm#koornwinderDefinition","koornwinderNonsymmetric":"koornwinder.htm#koornwinderNonsymmetric","koornwinderPolynomial":"koornwinder.htm#koornwinderPolynomial","koornwinderPolynomialDefinition":"koornwinder.htm#koornwinderPolynomialDefinition","koornwinderSpecializations":"koornwinder.htm#koornwinderSpecializations","kostantKumarModule":"representationTheory.htm#kostantKumarModule","kostantMultiplicityFormula":"kostkaFoulkes.htm#kostantMultiplicityFormula","kostantPartitionFunction":"kostkaFoulkes.htm#kostantPartitionFunction","kostantPartitionFunctionDefinition":"kostkaFoulkes.htm#kostantPartitionFunctionDefinition","kostantPartitionFunctionGeneralType":"kostkaFoulkes.htm#kostantPartitionFunctionGeneralType","kostantPartitionFunctionOtherTypes":"kostkaFoulkes.htm#kostantPartitionFunctionOtherTypes","kostkaCoefficient":"kostkaFoulkes.htm#kostkaCoefficient","kostkaCoefficients":"kostkaFoulkes.htm#kostkaCoefficients","kostkaFoulkes":"kostkaFoulkes.htm#kostkaFoulkes","kostkaFoulkesAtoms":"kostkaFoulkes.htm#kostkaFoulkesAtoms","kostkaFoulkesGeneralizations":"kostkaFoulkes.htm#kostkaFoulkesGeneralizations","kostkaFoulkesParabolic":"kostkaFoulkes.htm#kostkaFoulkesParabolic","kostkaFoulkesPolynomial":"kostkaFoulkes.htm#kostkaFoulkesPolynomial","kostkaFoulkesPolynomials":"hallLittlewood.htm#kostkaFoulkesPolynomials","kostkaFoulkesRect":"kostkaFoulkes.htm#kostkaFoulkesRect","kostkaFromContingency":"kostkaFoulkes.htm#kostkaFromContingency","kostkaFromKostant":"kostkaFoulkes.htm#kostkaFromKostant","kostkaFromLR":"kostkaFoulkes.htm#kostkaFromLR","kostkaIdentities":"kostkaFoulkes.htm#kostkaIdentities","kostkaInverseModel":"standardSymmetricFunctions.htm#kostkaInverseModel","kostkaInverseRecursion":"kostkaFoulkes.htm#kostkaInverseRecursion","kostkaRecursion":"kostkaFoulkes.htm#kostkaRecursion","kostkaStretched":"kostkaFoulkes.htm#kostkaStretched","kpromotion":"tableauOperators.htm#kpromotion","kromaticSymmetric":"kromaticSymmetric.htm#kromaticSymmetric","kroneckerCoefficient":"schur.htm#kroneckerCoefficient","kurtzTheorem":"realRooted.htm#kurtzTheorem","labeledDyckPathDefinition":"parking-functions.htm#labeledDyckPathDefinition","labeledPPartition":"pPartitions.htm#labeledPPartition","labeledPoset":"posets.htm#labeledPoset","laceMatrix":"realRootedInterlacing.htm#laceMatrix","lagrangeInversion":"lagrangeInversion.htm#lagrangeInversion","laguerrePolyaClass":"stablePolynomials.htm#laguerrePolyaClass","lah":"lah.htm#lah","lahForestBasis":"lah.htm#lahForestBasis","lambdaArray":"rsk.htm#lambdaArray","lambdaUnimodalSubset":"gessel.htm#lambdaUnimodalSubset","laminarFamily":"matroids.htm#laminarFamily","laminarMatroid":"matroids.htm#laminarMatroid","laminarMatroids":"matroids.htm#laminarMatroids","laplaceBeltramiOperator":"jack.htm#laplaceBeltramiOperator","laplacianMatchingPolynomial":"realRootedGraphs.htm#laplacianMatchingPolynomial","lascoux":"lascoux.htm#lascoux","lascouxAtom":"lascoux.htm#lascouxAtom","lascouxAtomVertexModel":"lascoux.htm#lascouxAtomVertexModel","lascouxDefinition":"lascoux.htm#lascouxDefinition","lascouxPolynomialDefinition":"lascoux.htm#lascouxPolynomialDefinition","lascouxProductAtomCounterexamples":"lascoux.htm#lascouxProductAtomCounterexamples","lascouxProperties":"lascoux.htm#lascouxProperties","lascouxSchutzenbergerInvolution":"crystals.htm#lascouxSchutzenbergerInvolution","lascouxSetValuedTableau":"lascoux.htm#lascouxSetValuedTableau","lascouxTableauhedron":"key.htm#lascouxTableauhedron","lascouxTableauhedronDefinition":"key.htm#lascouxTableauhedronDefinition","latticeModel":"latticeModel.htm#latticeModel","latticeModelList":"latticeModel.htm#latticeModelList","latticeModelVertex":"latticeModel.htm#latticeModelVertex","latticeOfFlats":"matroids.htm#latticeOfFlats","latticePathMatroidDefinition":"lattice-path-matroids.htm#latticePathMatroidDefinition","latticePathMatroids":"lattice-path-matroids.htm#latticePathMatroids","latticePolytope":"polytopes.htm#latticePolytope","latticePoset":"posets.htm#latticePoset","latticeWord":"gtpatterns.htm#latticeWord","laurentSymmetricFunctions":"standardSymmetricFunctions.htm#laurentSymmetricFunctions","laurentSymmetricPolynomial":"standardSymmetricFunctions.htm#laurentSymmetricPolynomial","lcNizPolynomial":"realRooted.htm#lcNizPolynomial","lectureHallTableau":"schurMisc.htm#lectureHallTableau","lehmerCode":"permutations.htm#lehmerCode","lehmerCodeDefinition":"permutations.htm#lehmerCodeDefinition","lgvLemma":"lgv-lemma.htm#lgvLemma","lindstromGesselViennotLemma":"lgv-lemma.htm#lindstromGesselViennotLemma","lineGraph":"realRootedGraphs.htm#lineGraph","linearExtension":"posets.htm#linearExtension","linearExtensions":"posets.htm#linearExtensions","linearToggle":"cspTableau.htm#linearToggle","littlewoodMap":"borderStripTableaux.htm#littlewoodMap","littlewoodRichardsonCoefficient":"schur.htm#littlewoodRichardsonCoefficient","littlewoodRichardsonFromKostant":"kostkaFoulkes.htm#littlewoodRichardsonFromKostant","littlewoodRichardsonModels":"littlewoodRichardson.htm#littlewoodRichardsonModels","littlewoodRichardsonTableaux":"littlewoodRichardson.htm#littlewoodRichardsonTableaux","littlewoodSchur":"superSymmetricSchur.htm#littlewoodSchur","lltCauchy":"llt.htm#lltCauchy","lltDefinitionRibbons":"llt.htm#lltDefinitionRibbons","lltDefinitionTuples":"llt.htm#lltDefinitionTuples","lltFactorial":"llt.htm#lltFactorial","lltFundamentalExpansion":"llt.htm#lltFundamentalExpansion","lltInversion":"llt.htm#lltInversion","lltMurnaghanNakayama":"llt.htm#lltMurnaghanNakayama","lltNablaHallLittlewoodCovers":"llt.htm#lltNablaHallLittlewoodCovers","lltPieri":"llt.htm#lltPieri","lltPolynomialDefinition":"llt.htm#lltPolynomialDefinition","lltProperties":"llt.htm#lltProperties","lltReadingOrder":"llt.htm#lltReadingOrder","lltSchurPositivity":"llt.htm#lltSchurPositivity","lltSuperSym":"llt.htm#lltSuperSym","lltUnicellularPointer":"llt.htm#lltUnicellularPointer","lltVertexModel":"llt.htm#lltVertexModel","lock":"assaf.htm#lock","lockPolynomialDefinition":"assaf.htm#lockPolynomialDefinition","lockTableau":"assaf.htm#lockTableau","logConcavePolynomial":"realRooted.htm#logConcavePolynomial","loopSymmetricFunctions":"loopSchur.htm#loopSymmetricFunctions","lorentzianPolynomial":"lorentzianPolynomials.htm#lorentzianPolynomial","lorentzianSkewSchur":"lorentzianPolynomials.htm#lorentzianSkewSchur","lowerHookLength":"jack.htm#lowerHookLength","lrCharge":"littlewoodRichardson.htm#lrCharge","lrCoquereauxZuber":"littlewoodRichardson.htm#lrCoquereauxZuber","lrExplicitFormula":"littlewoodRichardson.htm#lrExplicitFormula","lrFitsSkewShape":"littlewoodRichardson.htm#lrFitsSkewShape","lrIdentities":"littlewoodRichardson.htm#lrIdentities","lrLatticeWord":"littlewoodRichardson.htm#lrLatticeWord","lrMolevSagan":"littlewoodRichardson.htm#lrMolevSagan","lrSaturationTheorem":"gtpatterns.htm#lrSaturationTheorem","lrShrivastava":"littlewoodRichardson.htm#lrShrivastava","lrStretched":"littlewoodRichardson.htm#lrStretched","lrTableaux":"littlewoodRichardson.htm#lrTableaux","lrYamanouchi":"littlewoodRichardson.htm#lrYamanouchi","lrcalc":"littlewoodRichardson.htm#lrcalc","lyndonCSP":"cyclic-sieving.htm#lyndonCSP","lyndonCSPII":"cyclic-sieving.htm#lyndonCSPII","lyndonSymmetric":"cycleIndexPolynomial.htm#lyndonSymmetric","lyndonSymmetricFunctionDefinition":"cycleIndexPolynomial.htm#lyndonSymmetricFunctionDefinition","mConvexSet":"newtonPolytopes.htm#mConvexSet","mSymmetricFunction":"macdonaldP.htm#mSymmetricFunction","maWangDerivativeCriterion":"realRootedInterlacing.htm#maWangDerivativeCriterion","macMahonQCatalan":"q-analogs.htm#macMahonQCatalan","macdonaldCharacter":"macdonaldP.htm#macdonaldCharacter","macdonaldDifferenceOperator":"macdonaldP.htm#macdonaldDifferenceOperator","macdonaldE":"macdonaldE.htm#macdonaldE","macdonaldEMonks":"macdonaldE.htm#macdonaldEMonks","macdonaldEPermShape":"macdonaldEperm.htm#macdonaldEPermShape","macdonaldEQuasi":"macdonaldEperm.htm#macdonaldEQuasi","macdonaldERecurrences":"macdonaldE.htm#macdonaldERecurrences","macdonaldEperm":"macdonaldEperm.htm#macdonaldEperm","macdonaldEpermDefinition":"macdonaldEperm.htm#macdonaldEpermDefinition","macdonaldEpermProperties":"macdonaldEperm.htm#macdonaldEpermProperties","macdonaldEspec":"macdonaldE.htm#macdonaldEspec","macdonaldFillingArm":"macdonaldEperm.htm#macdonaldFillingArm","macdonaldFillingDescent":"macdonaldEperm.htm#macdonaldFillingDescent","macdonaldFillingInversionNumber":"macdonaldEperm.htm#macdonaldFillingInversionNumber","macdonaldFillingLeg":"macdonaldEperm.htm#macdonaldFillingLeg","macdonaldFillingMajorIndex":"macdonaldEperm.htm#macdonaldFillingMajorIndex","macdonaldH":"macdonaldH.htm#macdonaldH","macdonaldHCombinatorialFormula":"macdonaldH.htm#macdonaldHCombinatorialFormula","macdonaldHCombinatorialFormula2":"macdonaldH.htm#macdonaldHCombinatorialFormula2","macdonaldHCumulants":"macdonaldH.htm#macdonaldHCumulants","macdonaldHDefinitionViaMacdonaldJ":"macdonaldH.htm#macdonaldHDefinitionViaMacdonaldJ","macdonaldHNonCommutative":"macdonaldH.htm#macdonaldHNonCommutative","macdonaldHProperties":"macdonaldH.htm#macdonaldHProperties","macdonaldHSchurPositivity":"macdonaldH.htm#macdonaldHSchurPositivity","macdonaldHSkew":"macdonaldH.htm#macdonaldHSkew","macdonaldHUniqueDefinition":"macdonaldH.htm#macdonaldHUniqueDefinition","macdonaldHqtKostka":"macdonaldH.htm#macdonaldHqtKostka","macdonaldHstretching":"macdonaldH.htm#macdonaldHstretching","macdonaldInnerProduct":"macdonaldP.htm#macdonaldInnerProduct","macdonaldJ":"macdonaldP.htm#macdonaldJ","macdonaldJHaglundsConjecture":"macdonaldP.htm#macdonaldJHaglundsConjecture","macdonaldJQuasi":"macdonaldP.htm#macdonaldJQuasi","macdonaldJqtKostka":"macdonaldP.htm#macdonaldJqtKostka","macdonaldLittlewoodRichardsonCoefficient":"macdonaldP.htm#macdonaldLittlewoodRichardsonCoefficient","macdonaldP":"macdonaldP.htm#macdonaldP","macdonaldPCreationCharacters":"macdonaldP.htm#macdonaldPCreationCharacters","macdonaldPEigenvectors":"macdonaldP.htm#macdonaldPEigenvectors","macdonaldPFromE":"macdonaldP.htm#macdonaldPFromE","macdonaldPMulti":"macdonaldP.htm#macdonaldPMulti","macdonaldPProperties":"macdonaldP.htm#macdonaldPProperties","macdonaldPTableauFormula":"macdonaldP.htm#macdonaldPTableauFormula","macdonaldPUniqueDefinition":"macdonaldP.htm#macdonaldPUniqueDefinition","macdonaldPWreath":"macdonaldP.htm#macdonaldPWreath","macdonaldPi":"macdonaldPinterpolation.htm#macdonaldPi","macdonaldPiBC":"macdonaldPinterpolation.htm#macdonaldPiBC","macdonaldPiRSSYTFormula":"macdonaldPinterpolation.htm#macdonaldPiRSSYTFormula","macdonaldPolynomialDefinition":"macdonaldP.htm#macdonaldPolynomialDefinition","macdonaldPqq2":"macdonaldP.htm#macdonaldPqq2","macdonaldSeventhVariationSchur":"schubertVariations.htm#macdonaldSeventhVariationSchur","magicPositiveEhrhart":"polytopes.htm#magicPositiveEhrhart","magicPositiveEhrhartDefinition":"polytopes.htm#magicPositiveEhrhartDefinition","magicPositivePolynomial":"polytopes.htm#magicPositivePolynomial","mahonianStatistic":"q-analogs.htm#mahonianStatistic","majorIndex":"preliminaries.htm#majorIndex","mapsOnSymmetricFunctions":"symmetricFunctionsSpace.htm#mapsOnSymmetricFunctions","markedChainOrderPolytope":"gtpatterns.htm#markedChainOrderPolytope","markedChainPolytope":"gtpatterns.htm#markedChainPolytope","markedOrderPolytope":"polytopes.htm#markedOrderPolytope","markedOrderPolytopeDefinition":"polytopes.htm#markedOrderPolytopeDefinition","markedOrderPolytopePositivity":"polytopes.htm#markedOrderPolytopePositivity","matchingFieldPolytope":"gtpatterns.htm#matchingFieldPolytope","matchingMatroid":"matroids.htm#matchingMatroid","matchingPolynomial":"realRootedGraphs.htm#matchingPolynomial","matricubes":"matroids.htm#matricubes","matrixPreservesInterlacingSequences":"realRootedInterlacing.htm#matrixPreservesInterlacingSequences","matroid":"matroids.htm#matroid","matroidAdjacentPages":"matroids.htm#matroidAdjacentPages","matroidApplicationsMisc":"matroids.htm#matroidApplicationsMisc","matroidBasePolytope":"matroids.htm#matroidBasePolytope","matroidBases":"matroids.htm#matroidBases","matroidBasis":"matroids.htm#matroidBasis","matroidCharacteristicPolynomial":"matroids.htm#matroidCharacteristicPolynomial","matroidChowPolynomials":"matroids.htm#matroidChowPolynomials","matroidChowRing":"matroids.htm#matroidChowRing","matroidColoop":"matroids.htm#matroidColoop","matroidContraction":"matroids.htm#matroidContraction","matroidDefinition":"matroids.htm#matroidDefinition","matroidDeletion":"matroids.htm#matroidDeletion","matroidDual":"matroids.htm#matroidDual","matroidDuality":"matroids.htm#matroidDuality","matroidFamilies":"matroids.htm#matroidFamilies","matroidFlat":"matroids.htm#matroidFlat","matroidFlats":"matroids.htm#matroidFlats","matroidGraphic":"matroids.htm#matroidGraphic","matroidIndependentSets":"matroids.htm#matroidIndependentSets","matroidIsomorphism":"matroids.htm#matroidIsomorphism","matroidKazhdanLusztigPolynomial":"matroids.htm#matroidKazhdanLusztigPolynomial","matroidLatticePathSummary":"matroids.htm#matroidLatticePathSummary","matroidLinks":"matroids.htm#matroidLinks","matroidLogConcavityStability":"matroids.htm#matroidLogConcavityStability","matroidLoop":"matroids.htm#matroidLoop","matroidLoopsColoops":"matroids.htm#matroidLoopsColoops","matroidMatching":"matroids.htm#matroidMatching","matroidMinors":"matroids.htm#matroidMinors","matroidOperations":"matroids.htm#matroidOperations","matroidOptimization":"matroids.htm#matroidOptimization","matroidPolytopes":"matroids.htm#matroidPolytopes","matroidPositroidSummary":"matroids.htm#matroidPositroidSummary","matroidRank":"matroids.htm#matroidRank","matroidRankDefinition":"matroids.htm#matroidRankDefinition","matroidRealizability":"matroids.htm#matroidRealizability","matroidRegular":"matroids.htm#matroidRegular","matroidRestriction":"matroids.htm#matroidRestriction","matroidSnapperPolynomials":"matroids.htm#matroidSnapperPolynomials","matroidUniform":"matroids.htm#matroidUniform","matroidValuativeInvariants":"matroids.htm#matroidValuativeInvariants","matroidVariantsAdjacentClasses":"matroids.htm#matroidVariantsAdjacentClasses","metaPapers":"todo-list.htm#metaPapers","metaWriteFamilies":"todo-list.htm#metaWriteFamilies","metaWriteTopics":"todo-list.htm#metaWriteTopics","metaplecticSymmetricFunction":"llt.htm#metaplecticSymmetricFunction","miscLinks":"resources.htm#miscLinks","mistakesColon":"common-mistakes.htm#mistakesColon","mistakesEquationMode":"common-mistakes.htm#mistakesEquationMode","mistakesIntro":"common-mistakes.htm#mistakesIntro","mistakesSentenceStart":"common-mistakes.htm#mistakesSentenceStart","mistakesSymbols":"common-mistakes.htm#mistakesSymbols","mistakesType":"common-mistakes.htm#mistakesType","mistakesTypesEqual":"common-mistakes.htm#mistakesTypesEqual","mistakesTypesIntro":"common-mistakes.htm#mistakesTypesIntro","mistakesTypesLimits":"common-mistakes.htm#mistakesTypesLimits","mixedCharacteristicPolynomial":"stablePolynomials.htm#mixedCharacteristicPolynomial","mixedCharacteristicPolynomialBound":"stablePolynomials.htm#mixedCharacteristicPolynomialBound","mixedCharacteristicPolynomialDefinition":"stablePolynomials.htm#mixedCharacteristicPolynomialDefinition","mixedRSK":"rsk.htm#mixedRSK","mnHookSchurFunction":"superSymmetricSchur.htm#mnHookSchurFunction","mnRecursion":"murnaghanNakayama.htm#mnRecursion","mobiusSymmetricFunction":"weightedBondSymmetric.htm#mobiusSymmetricFunction","modifiedHallLittlewoodPolynomial":"hallLittlewood.htm#modifiedHallLittlewoodPolynomial","modifiedKostkaFoulkesPolynomial":"kostkaFoulkes.htm#modifiedKostkaFoulkesPolynomial","modifiedKostkaFoulkesPolynomials":"hallLittlewood.htm#modifiedKostkaFoulkesPolynomials","modifiedMacdonaldKostkaPolynomial":"macdonaldH.htm#modifiedMacdonaldKostkaPolynomial","modifiedMacdonaldPolynomial":"macdonaldH.htm#modifiedMacdonaldPolynomial","modularSchur":"petrie.htm#modularSchur","modularSchurDefinition":"petrie.htm#modularSchurDefinition","monomial":"standardSymmetricFunctions.htm#monomial","monomialQuasisymmetricFunctionDefinition":"standardQuasiSymmetricFunctions.htm#monomialQuasisymmetricFunctionDefinition","monomialSlidePolynomialDefinition":"assaf.htm#monomialSlidePolynomialDefinition","monotoneTriangle":"gtpatterns.htm#monotoneTriangle","motzkinPolynomialsRealRooted":"realRootedCatalan.htm#motzkinPolynomialsRealRooted","multiAffineEdgeMatchingPolynomial":"stablePolynomials.htm#multiAffineEdgeMatchingPolynomial","multiAffinePolynomial":"stablePolynomials.htm#multiAffinePolynomial","multiAffineVertexMatchingPolynomial":"stablePolynomials.htm#multiAffineVertexMatchingPolynomial","multiEulerianPolynomial":"realRootedWords.htm#multiEulerianPolynomial","multiMacdonaldPolynomial":"macdonaldP.htm#multiMacdonaldPolynomial","multiQuasisymmetricFunction":"standardQuasiSymmetricFunctions.htm#multiQuasisymmetricFunction","multiSchur":"superSymmetricSchur.htm#multiSchur","multiSchurFunction":"superSymmetricSchur.htm#multiSchurFunction","multiSymmetricSchurFunction":"standardSymmetricFunctions.htm#multiSymmetricSchurFunction","multilineQueueMacdonald":"macdonaldE.htm#multilineQueueMacdonald","multipleSchurSeries":"schurZeta.htm#multipleSchurSeries","multipleSchurSeriesDefinition":"schurZeta.htm#multipleSchurSeriesDefinition","multiplierSequenceDefinition":"realRooted.htm#multiplierSequenceDefinition","multiplierSequences":"realRooted.htm#multiplierSequences","multisetRSK":"rsk.htm#multisetRSK","multivariateIndependencePolynomial":"stablePolynomials.htm#multivariateIndependencePolynomial","murnaghanNakayamaRule":"murnaghanNakayama.htm#murnaghanNakayamaRule","murnaghanNakayamaRuleDefinition":"murnaghanNakayama.htm#murnaghanNakayamaRuleDefinition","murnaghanNakayamaVariants":"murnaghanNakayama.htm#murnaghanNakayamaVariants","myEquationsBSD":"my-equations.htm#myEquationsBSD","myEquationsCatalan":"my-equations.htm#myEquationsCatalan","myEquationsCircularDyck":"my-equations.htm#myEquationsCircularDyck","myEquationsJackKostka":"my-equations.htm#myEquationsJackKostka","myEquationsLltE":"my-equations.htm#myEquationsLltE","nablaHallLittlewoodBGHT":"shuffleAndDelta.htm#nablaHallLittlewoodBGHT","nablaOperator":"diagonalHarmonics.htm#nablaOperator","nagelsbachKostkaIdentity":"schur.htm#nagelsbachKostkaIdentity","namedPermutations":"permutationFamilies.htm#namedPermutations","narayanaNumberCSP":"cspCatalan.htm#narayanaNumberCSP","narayanaNumbers":"diagonalHarmonics.htm#narayanaNumbers","narayanaPolynomialFamiliesRealRooted":"realRootedCatalan.htm#narayanaPolynomialFamiliesRealRooted","narayanaTransformationDefinition":"realRooted.htm#narayanaTransformationDefinition","narayanaTransformations":"realRooted.htm#narayanaTransformations","naturalLabeling":"posets.htm#naturalLabeling","ncSchur":"ncSchur.htm#ncSchur","neggersStanleyConjecture":"realRooted.htm#neggersStanleyConjecture","neggersStanleyConjectureDefinition":"realRooted.htm#neggersStanleyConjectureDefinition","newellLittlewood":"littlewoodRichardson.htm#newellLittlewood","newellLittlewoodNumbersDefinition":"littlewoodRichardson.htm#newellLittlewoodNumbersDefinition","newtonInequalities":"realRooted.htm#newtonInequalities","newtonLorentzianBridge":"newtonPolytopes.htm#newtonLorentzianBridge","newtonPolytope":"newtonPolytopes.htm#newtonPolytope","newtonPolytopes":"newtonPolytopes.htm#newtonPolytopes","nonAttackingFilling":"jack.htm#nonAttackingFilling","nonAttackingFillingMacdonald":"macdonaldEperm.htm#nonAttackingFillingMacdonald","nonAttackingRookPlacement":"lattice-path-matroids.htm#nonAttackingRookPlacement","nonComDualBases":"nonCommutativeFunctions.htm#nonComDualBases","nonComFunctions":"nonCommutativeFunctions.htm#nonComFunctions","nonComHomogeneous":"nonCommutativeFunctions.htm#nonComHomogeneous","nonComMNrule":"nonCommutativeFunctions.htm#nonComMNrule","nonComPowerPhi":"nonCommutativeFunctions.htm#nonComPowerPhi","nonComPowerPsi":"nonCommutativeFunctions.htm#nonComPowerPsi","nonComRibbonBasis":"nonCommutativeFunctions.htm#nonComRibbonBasis","nonComSchur":"nonCommutativeFunctions.htm#nonComSchur","nonCommutativeShiftedSymmetricFunctions":"nonCommutativeShifted.htm#nonCommutativeShiftedSymmetricFunctions","nonCrossingMatchingCSP":"cspCatalan.htm#nonCrossingMatchingCSP","nonCrossingOneTwoConfiguration":"cspCatalan.htm#nonCrossingOneTwoConfiguration","nonCrossingPartitionCSP":"cspCatalan.htm#nonCrossingPartitionCSP","nonNestingRookPlacement":"lattice-path-matroids.htm#nonNestingRookPlacement","nonNestingRookPolynomial":"lattice-path-matroids.htm#nonNestingRookPolynomial","nonSymmetricElementary":"macdonaldE.htm#nonSymmetricElementary","nonSymmetricElementaryPolynomial":"macdonaldE.htm#nonSymmetricElementaryPolynomial","nonSymmetricMacdonaldPolynomial":"macdonaldE.htm#nonSymmetricMacdonaldPolynomial","nonSymmetricPermutedBasementMacdonaldPolynomial":"macdonaldEperm.htm#nonSymmetricPermutedBasementMacdonaldPolynomial","nonSymmetricQWhittakerPolynomial":"macdonaldE.htm#nonSymmetricQWhittakerPolynomial","noncommutativeCompleteHomogeneousFunction":"nonCommutativeFunctions.htm#noncommutativeCompleteHomogeneousFunction","noncommutativeDeterminant":"ncSchur.htm#noncommutativeDeterminant","noncommutativeForgetfulMorphism":"nonCommutativeFunctions.htm#noncommutativeForgetfulMorphism","noncommutativePowerSumBasis":"nonCommutativeFunctions.htm#noncommutativePowerSumBasis","noncommutativeSchurFunctionDefinition":"nonCommutativeFunctions.htm#noncommutativeSchurFunctionDefinition","noncommutativeSymmetricFunctionDefinition":"nonCommutativeFunctions.htm#noncommutativeSymmetricFunctionDefinition","noncrossingChowPolynomial":"realRootedGraphs.htm#noncrossingChowPolynomial","nonintersectingPaths":"lgv-lemma.htm#nonintersectingPaths","nonsymmetricRSK":"rsk.htm#nonsymmetricRSK","notationAffinePermutations":"permutationGeneralizations.htm#notationAffinePermutations","notationPermutations":"preliminaries.htm#notationPermutations","npCompleteGames":"np-complete-games.htm#npCompleteGames","obreschkoffDedieu":"realRootedInterlacing.htm#obreschkoffDedieu","oddSchur":"oddSchur.htm#oddSchur","oeisProofA152947":"oeisA152947.htm#oeisProofA152947","oeisProofA189912":"oeisA189912.htm#oeisProofA189912","oeisProofA358628":"oeisA358628.htm#oeisProofA358628","oeisProofs":"oeisProofs.htm#oeisProofs","omegaInvolutionDefinition":"symmetricFunctionsSpace.htm#omegaInvolutionDefinition","operatorPreservesInterlacing":"realRootedInterlacing.htm#operatorPreservesInterlacing","orbitalHarmonics":"cyclic-sieving.htm#orbitalHarmonics","orderFilter":"posets.htm#orderFilter","orderIdeal":"posets.htm#orderIdeal","orderIdealToggle":"rowmotionPromotion.htm#orderIdealToggle","orderIdealToggles":"rowmotionPromotion.htm#orderIdealToggles","orderPolynomial":"pPartitions.htm#orderPolynomial","orderPolynomials":"pPartitions.htm#orderPolynomials","orderPolytope":"polytopes.htm#orderPolytope","orderPolytopeDefinition":"polytopes.htm#orderPolytopeDefinition","orderPreservingSurjection":"pPartitions.htm#orderPreservingSurjection","orderPreservingSurjectionType":"unicellular-llt.htm#orderPreservingSurjectionType","orderQuasisymmetricFunction":"chromaticQuasisymmetric.htm#orderQuasisymmetricFunction","ordinalSumPoset":"realRooted.htm#ordinalSumPoset","orientedGraph":"chromaticQuasisymmetric.htm#orientedGraph","orientedMatroid":"matroids.htm#orientedMatroid","orthogonalSchurFunction":"schurSymplecticOrthogonal.htm#orthogonalSchurFunction","oscillatingTableau":"schurSymplecticOrthogonal.htm#oscillatingTableau","otherRSK":"rsk.htm#otherRSK","outsideDecomposition":"schur.htm#outsideDecomposition","pCore":"preliminaries.htm#pCore","pEulerianPolynomial":"posets.htm#pEulerianPolynomial","pExpansionTutte":"tutteSymmetric.htm#pExpansionTutte","pGrothendieck":"pGrothendieck.htm#pGrothendieck","pPartition":"pPartitions.htm#pPartition","pPartitionDefinition":"pPartitions.htm#pPartitionDefinition","pPartitionGeneratingFunction":"pPartitions.htm#pPartitionGeneratingFunction","pPartitionInGessel":"pPartitions.htm#pPartitionInGessel","pPartitionLorentzian":"pPartitions.htm#pPartitionLorentzian","pPartitionMNrule":"pPartitions.htm#pPartitionMNrule","pPartitionPsiExp":"pPartitions.htm#pPartitionPsiExp","pPartitionsFlagged":"pPartitions.htm#pPartitionsFlagged","palindromicPolynomial":"gammaPositivity.htm#palindromicPolynomial","parabolicKostkaPolynomial":"kostkaFoulkes.htm#parabolicKostkaPolynomial","parallelogramPolyomino":"diagonalHarmonics.htm#parallelogramPolyomino","parking":"parkingFunctionSymmetric.htm#parking","parkingFunctionDefinition":"parking-functions.htm#parkingFunctionDefinition","parkingFunctionDescentsRealRooted":"realRootedWords.htm#parkingFunctionDescentsRealRooted","parkingFunctionPolytope":"polytopes.htm#parkingFunctionPolytope","parkingFunctionPolytopeDefinition":"polytopes.htm#parkingFunctionPolytopeDefinition","parkingFunctionVariations":"parking-functions.htm#parkingFunctionVariations","parkingFunctions":"parking-functions.htm#parkingFunctions","parkingFunctionsBrief":"combinatorialObjects.htm#parkingFunctionsBrief","parkingFunctionsHyperplanes":"parking-functions.htm#parkingFunctionsHyperplanes","parkingFunctionsLabeledDyck":"parking-functions.htm#parkingFunctionsLabeledDyck","parkingFunctionsLabeledTrees":"parking-functions.htm#parkingFunctionsLabeledTrees","parkingPreference":"shuffleAndDelta.htm#parkingPreference","partialFlagVarietyDefinition":"varieties.htm#partialFlagVarietyDefinition","partialOrderingsOnPartitions":"preliminaries.htm#partialOrderingsOnPartitions","partialTransversal":"matroids.htm#partialTransversal","partiallySymmetricMacdonaldPolynomial":"macdonaldP.htm#partiallySymmetricMacdonaldPolynomial","partitionCores":"preliminaries.htm#partitionCores","partitionDefinition":"preliminaries.htm#partitionDefinition","partitionMatroid":"matroids.htm#partitionMatroid","partitionMatroids":"matroids.htm#partitionMatroids","partitionQuotient":"borderStripTableaux.htm#partitionQuotient","partitionType":"preliminaries.htm#partitionType","partyAlgebra":"plethysm.htm#partyAlgebra","pathGraphToricPromotion":"cspWord.htm#pathGraphToricPromotion","patternAvoidanceClasses":"permutationPatterns.htm#patternAvoidanceClasses","patternsAndRSK":"rsk.htm#patternsAndRSK","pavingMatroid":"matroids.htm#pavingMatroid","peakAbscissaSum":"schurMisc.htm#peakAbscissaSum","peakGeneratingPolynomial":"realRootedWords.htm#peakGeneratingPolynomial","peakQSym":"peakQuasisymmetric.htm#peakQSym","peakRunDerangementPolynomials":"realRootedWords.htm#peakRunDerangementPolynomials","peakRunPolynomialsRealRooted":"realRootedWords.htm#peakRunPolynomialsRealRooted","peakYoungQuasisymmetricSchurFunction":"qsymSchur.htm#peakYoungQuasisymmetricSchurFunction","peakYqSchur":"qsymSchur.htm#peakYqSchur","perfectMatchingPolytope":"polytopes.htm#perfectMatchingPolytope","perfectMatchingPolytopeDefinition":"polytopes.htm#perfectMatchingPolytopeDefinition","periodCollapse":"gtpatterns.htm#periodCollapse","permSeparable":"permutationFamilies.htm#permSeparable","permUpDown":"permutationFamilies.htm#permUpDown","permutationChargeStatistic":"preliminaries.htm#permutationChargeStatistic","permutationCochargeStatistic":"preliminaries.htm#permutationCochargeStatistic","permutationFamilyDescriptions":"permutationFamilies.htm#permutationFamilyDescriptions","permutationFamilyEnumeration":"permutationFamilies.htm#permutationFamilyEnumeration","permutationLengthDefinition":"permutations.htm#permutationLengthDefinition","permutationMatrix":"permutations.htm#permutationMatrix","permutationMatrixDefinition":"permutations.htm#permutationMatrixDefinition","permutationPattern":"permutationPatterns.htm#permutationPattern","permutationPeak":"gammaPositivity.htm#permutationPeak","permutationsCayley":"permutationGeneralizations.htm#permutationsCayley","permutationsColored":"permutationGeneralizations.htm#permutationsColored","permutationsStirling":"permutationGeneralizations.htm#permutationsStirling","permutationsTypeB":"permutationGeneralizations.htm#permutationsTypeB","permutedBasementAtom":"macdonaldEperm.htm#permutedBasementAtom","permutedBasementAtomDefinition":"macdonaldEperm.htm#permutedBasementAtomDefinition","permutedBasementFillingDiagrams":"macdonaldEperm.htm#permutedBasementFillingDiagrams","permutedBasementFillingStatistics":"macdonaldEperm.htm#permutedBasementFillingStatistics","permutedBasementMacdonaldConventions":"macdonaldEperm.htm#permutedBasementMacdonaldConventions","permutedBasementMacdonaldExample":"macdonaldEperm.htm#permutedBasementMacdonaldExample","permutedBasementMacdonaldFormula":"macdonaldEperm.htm#permutedBasementMacdonaldFormula","permutedBasementMacdonaldPolynomial":"macdonaldEperm.htm#permutedBasementMacdonaldPolynomial","permutedBasementTAtom":"macdonaldEperm.htm#permutedBasementTAtom","permutedBasementTAtomDefinition":"macdonaldEperm.htm#permutedBasementTAtomDefinition","permutohedron":"polytopes.htm#permutohedron","permutohedronDefinition":"polytopes.htm#permutohedronDefinition","permutoricPromotion":"cspWord.htm#permutoricPromotion","personalMinorProblems":"various-research.htm#personalMinorProblems","petersonVarietyDefinition":"varieties.htm#petersonVarietyDefinition","petrie":"petrie.htm#petrie","petrieCombinatorial":"petrie.htm#petrieCombinatorial","petrieDeterminant":"petrie.htm#petrieDeterminant","petrieHopf":"petrie.htm#petrieHopf","petrieInSchur":"petrie.htm#petrieInSchur","petrieMN":"petrie.htm#petrieMN","petrieNabla":"petrie.htm#petrieNabla","petriePrincipal":"petrie.htm#petriePrincipal","petrieRecursion":"petrie.htm#petrieRecursion","petrieSymmetricFunctionDefinition":"petrie.htm#petrieSymmetricFunctionDefinition","pfConvolution":"polyaFrequency.htm#pfConvolution","pfRiordanArrayCriterion":"polyaFrequency.htm#pfRiordanArrayCriterion","pfRiordanArrays":"polyaFrequency.htm#pfRiordanArrays","pfSchurJacobiTrudi":"polyaFrequency.htm#pfSchurJacobiTrudi","pfTwoLogConcavity":"polyaFrequency.htm#pfTwoLogConcavity","pfaffian":"touchardRiordan.htm#pfaffian","pfaffianDefinition":"touchardRiordan.htm#pfaffianDefinition","pieriPetrie":"petrie.htm#pieriPetrie","pipeDream":"schubert.htm#pipeDream","pitmanStanleyMagicPositivity":"polytopes.htm#pitmanStanleyMagicPositivity","pitmanStanleyPolytope":"polytopes.htm#pitmanStanleyPolytope","plabicGraphDefinition":"positroids.htm#plabicGraphDefinition","placticMonoid":"rsk.htm#placticMonoid","planePartitions":"planePartitions.htm#planePartitions","plethysm":"plethysm.htm#plethysm","plethysmComputation":"plethysm.htm#plethysmComputation","plethysmDefinition":"plethysm.htm#plethysmDefinition","plethysmFoulkesConjecture":"plethysm.htm#plethysmFoulkesConjecture","plethysmGLn":"plethysm.htm#plethysmGLn","plethysmIdentities":"plethysm.htm#plethysmIdentities","plethysmOperation":"plethysm.htm#plethysmOperation","plethysmOther":"plethysm.htm#plethysmOther","plethysmProblems":"plethysm.htm#plethysmProblems","plethysmWithSchur":"plethysm.htm#plethysmWithSchur","plethysticMNRule":"murnaghanNakayama.htm#plethysticMNRule","plethysticSubstitution":"plethysm.htm#plethysticSubstitution","pluckerCoordinate":"varieties.htm#pluckerCoordinate","pluckerEmbedding":"varieties.htm#pluckerEmbedding","pochhammerSymbol":"q-analogs.htm#pochhammerSymbol","poincarePolynomial":"kostkaFoulkes.htm#poincarePolynomial","polarizationDefinition":"stablePolynomials.htm#polarizationDefinition","polyaFrequency":"polyaFrequency.htm#polyaFrequency","polyaFrequencyExamples":"polyaFrequency.htm#polyaFrequencyExamples","polyaFrequencySequenceDefinition":"polyaFrequency.htm#polyaFrequencySequenceDefinition","polygonTriangulationEar":"cspCatalan.htm#polygonTriangulationEar","polymatroid":"matroids.htm#polymatroid","polymatroidRankFunction":"matroids.htm#polymatroidRankFunction","polymatroids":"matroids.htm#polymatroids","polynomialIndex":"polynomialList.htm#polynomialIndex","polynomialNewtonSums":"realRooted.htm#polynomialNewtonSums","polynomialRecurrences":"polynomialRecurrences.htm#polynomialRecurrences","polynomialRepresentation":"representationTheory.htm#polynomialRepresentation","polynomialRootIntro":"realRooted.htm#polynomialRootIntro","polynomialUnimodality":"realRooted.htm#polynomialUnimodality","polysymmetricFunctions":"standardSymmetricFunctions.htm#polysymmetricFunctions","polytabloidDefinition":"specht-modules.htm#polytabloidDefinition","polytabloids":"specht-modules.htm#polytabloids","polytopalMeasure":"polytopes.htm#polytopalMeasure","polytope":"polytopes.htm#polytope","polytopeFaceHPolynomial":"polytopes.htm#polytopeFaceHPolynomial","polytopeFacePolynomials":"polytopes.htm#polytopeFacePolynomials","polytopeFamilies":"polytopes.htm#polytopeFamilies","polytopeIDP":"polytopes.htm#polytopeIDP","polytopePapers":"polytopes.htm#polytopePapers","poset":"posets.htm#poset","posetChainPolynomials":"posets.htm#posetChainPolynomials","posetClasses":"posets.htm#posetClasses","posetDynamics":"posets.htm#posetDynamics","posetHookFormulas":"posets.htm#posetHookFormulas","posetIdeals":"posets.htm#posetIdeals","posetPolytopes":"posets.htm#posetPolytopes","posetPromotion":"rowmotionPromotion.htm#posetPromotion","posetRealRooted":"realRooted.htm#posetRealRooted","posetWidth":"posets.htm#posetWidth","posets":"posets.htm#posets","positiveInterlacingFamily":"realRootedInterlacing.htm#positiveInterlacingFamily","positivityProbabilities":"standardSymmetricFunctions.htm#positivityProbabilities","positroid":"positroids.htm#positroid","positroidDefinition":"positroids.htm#positroidDefinition","positroidExamples":"positroids.htm#positroidExamples","positroidGrassmannNecklace":"positroids.htm#positroidGrassmannNecklace","positroidNetworks":"positroids.htm#positroidNetworks","positroidOperations":"positroids.htm#positroidOperations","positroidPolytopeDefinition":"positroids.htm#positroidPolytopeDefinition","positroidPolytopes":"positroids.htm#positroidPolytopes","positroidProperties":"positroids.htm#positroidProperties","positroidVarietyDefinition":"positroids.htm#positroidVarietyDefinition","powerSum":"standardSymmetricFunctions.htm#powerSum","preferenceFunction":"shuffleAndDelta.htm#preferenceFunction","preferenceFunctionArea":"shuffleAndDelta.htm#preferenceFunctionArea","preferenceFunctionDefinition":"parking-functions.htm#preferenceFunctionDefinition","preferenceFunctionDinv":"shuffleAndDelta.htm#preferenceFunctionDinv","preferenceFunctionDinvStatistic":"parking-functions.htm#preferenceFunctionDinvStatistic","preferenceFunctionReadingWord":"parking-functions.htm#preferenceFunctionReadingWord","preferenceFunctions":"parking-functions.htm#preferenceFunctions","prelimEvalAtRootsOfUnity":"q-analogs.htm#prelimEvalAtRootsOfUnity","prelimFixedDescentSet":"q-analogs.htm#prelimFixedDescentSet","prelimPartitionStatistics":"preliminaries.htm#prelimPartitionStatistics","prelimPartitions":"preliminaries.htm#prelimPartitions","prelimQHook":"q-analogs.htm#prelimQHook","prelimQKostka":"q-analogs.htm#prelimQKostka","prelimQanalogsCatalan":"q-analogs.htm#prelimQanalogsCatalan","prelimQanalogsQLucas":"q-analogs.htm#prelimQanalogsQLucas","prelimQidentities":"q-analogs.htm#prelimQidentities","prelimTableaux":"preliminaries.htm#prelimTableaux","prelimTableauxAsPolytope":"preliminaries.htm#prelimTableauxAsPolytope","prelimTableauxSmartMap":"preliminaries.htm#prelimTableauxSmartMap","preliminaries":"preliminaries.htm#preliminaries","principalSpecialization":"standardSymmetricFunctions.htm#principalSpecialization","probabilisticRSK":"rsk.htm#probabilisticRSK","projectiveVarietyDefinition":"varieties.htm#projectiveVarietyDefinition","promotion":"tableauOperators.htm#promotion","promotionSorting":"tableauOperators.htm#promotionSorting","prop:kempfCase":"key.htm#prop:kempfCase","properColoring":"chromaticQuasisymmetric.htm#properColoring","properSplitting":"stablePolynomials.htm#properSplitting","qAnalogs":"q-analogs.htm#qAnalogs","qBinomialCoefficient":"q-analogs.htm#qBinomialCoefficient","qCatalanNumberCSP":"cspCatalan.htm#qCatalanNumberCSP","qChromaticPolynomial":"chromaticQuasisymmetric.htm#qChromaticPolynomial","qDerangementNumber":"q-analogs.htm#qDerangementNumber","qDerangementsAndShuffles":"q-analogs.htm#qDerangementsAndShuffles","qFactorial":"q-analogs.htm#qFactorial","qFussCatalan":"q-analogs.htm#qFussCatalan","qHookFormulaCSPTableau":"cspTableau.htm#qHookFormulaCSPTableau","qInequality":"various-research.htm#qInequality","qInteger":"q-analogs.htm#qInteger","qKrewerasNumber":"cspCatalan.htm#qKrewerasNumber","qMultinomialCoefficient":"q-analogs.htm#qMultinomialCoefficient","qNarayana":"q-analogs.htm#qNarayana","qNarayanaNumber":"q-analogs.htm#qNarayanaNumber","qPhi":"standardQuasiSymmetricFunctions.htm#qPhi","qPochhammerSymbol":"q-analogs.htm#qPochhammerSymbol","qPsi":"standardQuasiSymmetricFunctions.htm#qPsi","qRho":"standardQuasiSymmetricFunctions.htm#qRho","qSchur":"qsymSchur.htm#qSchur","qSchurQ":"qsymSchur.htm#qSchurQ","qShiftedFactorial":"q-analogs.htm#qShiftedFactorial","qWhittaker":"whittaker.htm#qWhittaker","qWhittakerFunctionDefinition":"whittaker.htm#qWhittakerFunctionDefinition","qWhittakerModels":"whittaker.htm#qWhittakerModels","qbinomial":"q-analogs.htm#qbinomial","qmonom":"standardQuasiSymmetricFunctions.htm#qmonom","qsymNSymDuality":"standardQuasiSymmetricFunctions.htm#qsymNSymDuality","qsymToricRichardsonGeometry":"standardQuasiSymmetricFunctions.htm#qsymToricRichardsonGeometry","qtCatalan":"qtCatalan.htm#qtCatalan","qtCatalanGeneralizations":"qtCatalan.htm#qtCatalanGeneralizations","qtChromaticSymmetricFunction":"chromaticEexpansion.htm#qtChromaticSymmetricFunction","qtKostkaPolynomial":"macdonaldP.htm#qtKostkaPolynomial","quantumDoubleGrothendieckPolynomial":"grothendieck.htm#quantumDoubleGrothendieckPolynomial","quantumKostka":"kostkaFoulkes.htm#quantumKostka","quantumKostkaCoefficient":"kostkaFoulkes.htm#quantumKostkaCoefficient","quantumMNRule":"murnaghanNakayama.htm#quantumMNRule","quasiCrystal":"crystals.htm#quasiCrystal","quasiCrystals":"crystals.htm#quasiCrystals","quasiKey":"kohnert.htm#quasiKey","quasiKeyPolynomial":"kohnert.htm#quasiKeyPolynomial","quasiKeyPolynomialDefinition":"kohnert.htm#quasiKeyPolynomialDefinition","quasiKohnertTableau":"kohnert.htm#quasiKohnertTableau","quasiLascouxBasis":"lascoux.htm#quasiLascouxBasis","quasiSchurFunctions":"qsymSchur.htm#quasiSchurFunctions","quasiStirlingDescentRealRooted":"realRootedSetPartitions.htm#quasiStirlingDescentRealRooted","quasiStirlingPermutation":"realRootedSetPartitions.htm#quasiStirlingPermutation","quasiSymmetricFunctions":"standardQuasiSymmetricFunctions.htm#quasiSymmetricFunctions","quasiYamanouchiKohnertTableau":"kohnert.htm#quasiYamanouchiKohnertTableau","quasisymmetricCharacteristic":"zeroHecke.htm#quasisymmetricCharacteristic","quasisymmetricFunctionDefinition":"standardQuasiSymmetricFunctions.htm#quasisymmetricFunctionDefinition","quasisymmetricMacdonaldEPolynomial":"macdonaldEperm.htm#quasisymmetricMacdonaldEPolynomial","quasisymmetricMacdonaldPolynomial":"macdonaldP.htm#quasisymmetricMacdonaldPolynomial","quasisymmetricPowerSumPsi":"standardQuasiSymmetricFunctions.htm#quasisymmetricPowerSumPsi","quasisymmetricSchurDefinition":"qsymSchur.htm#quasisymmetricSchurDefinition","quasisymmetricSchurFunction":"qsymSchur.htm#quasisymmetricSchurFunction","quasisymmetricSchurQFunction":"qsymSchur.htm#quasisymmetricSchurQFunction","quasisymmetricSchurTableaux":"qsymSchur.htm#quasisymmetricSchurTableaux","rainbowBasis":"matroids.htm#rainbowBasis","rankFunction":"matroids.htm#rankFunction","rankedPoset":"posets.htm#rankedPoset","rationalPolytope":"polytopes.htm#rationalPolytope","rationalQCatalanNumber":"q-analogs.htm#rationalQCatalanNumber","rationalRepresentation":"representationTheory.htm#rationalRepresentation","rationalShuffle":"shuffleAndDelta.htm#rationalShuffle","rationalShuffleTheorem":"shuffleAndDelta.htm#rationalShuffleTheorem","rcGraph":"schubert.htm#rcGraph","readingWord":"preliminaries.htm#readingWord","realMatrixRecursion":"realRootedInterlacing.htm#realMatrixRecursion","realRootedCatalan":"realRootedCatalan.htm#realRootedCatalan","realRootedChromaticW":"realRootedGraphs.htm#realRootedChromaticW","realRootedIndependence":"realRootedGraphs.htm#realRootedIndependence","realRootedMatching":"realRootedGraphs.htm#realRootedMatching","realRootedNormality":"realRooted.htm#realRootedNormality","realRootedPolynomial":"realRooted.htm#realRootedPolynomial","realRootedPreservers":"realRootedInterlacing.htm#realRootedPreservers","realRootedTableaux":"realRootedTableaux.htm#realRootedTableaux","realRootedTransformations":"realRooted.htm#realRootedTransformations","realRootedWords":"realRootedWords.htm#realRootedWords","realRootsChow":"realRootedGraphs.htm#realRootsChow","realizableMatroid":"matroids.htm#realizableMatroid","realrootedOpenProblems":"gammaPositivity.htm#realrootedOpenProblems","recordingTableau":"rsk.htm#recordingTableau","recurrenceAnnihilatingIdeal":"polynomialRecurrences.htm#recurrenceAnnihilatingIdeal","recurrenceCoefficientRestrictions":"polynomialRecurrences.htm#recurrenceCoefficientRestrictions","recurrenceComputation":"polynomialRecurrences.htm#recurrenceComputation","recurrenceFiniteVerification":"polynomialRecurrences.htm#recurrenceFiniteVerification","recurrenceGeneratingFunctions":"polynomialRecurrences.htm#recurrenceGeneratingFunctions","recurrencePropagation":"polynomialRecurrences.htm#recurrencePropagation","recurrenceRankCriterion":"polynomialRecurrences.htm#recurrenceRankCriterion","recurrenceSpectralModel":"polynomialRecurrences.htm#recurrenceSpectralModel","recurrenceTradeoffExample":"polynomialRecurrences.htm#recurrenceTradeoffExample","reducedExpressionDefinition":"permutations.htm#reducedExpressionDefinition","reducedKoganFace":"key.htm#reducedKoganFace","reducedWord":"permutations.htm#reducedWord","reducedWordDefinition":"permutations.htm#reducedWordDefinition","reducedWordExamples":"permutations.htm#reducedWordExamples","refinedDualStableGrothendieckPolynomial":"grothendieck.htm#refinedDualStableGrothendieckPolynomial","reflexivePolytope":"polytopes.htm#reflexivePolytope","reflexivePolytopeDefinition":"polytopes.htm#reflexivePolytopeDefinition","reflexivePolytopeMagicPositivity":"polytopes.htm#reflexivePolytopeMagicPositivity","regularMatroid":"matroids.htm#regularMatroid","regularPolytope":"polytopes.htm#regularPolytope","regularSemisimpleHessenbergVariety":"dot-action.htm#regularSemisimpleHessenbergVariety","regularUnimodularTriangulation":"matroids.htm#regularUnimodularTriangulation","reinerShimozonoConjecture":"key.htm#reinerShimozonoConjecture","reinerShimozonoCounterexamples":"key.htm#reinerShimozonoCounterexamples","repTheoryMore":"representationTheory.htm#repTheoryMore","representationCharacter":"representationTheory.htm#representationCharacter","representationCharacterDefinition":"representationTheory.htm#representationCharacterDefinition","representationDefinition":"representationTheory.htm#representationDefinition","representationDegree":"representationTheory.htm#representationDegree","representationGln":"representationTheory.htm#representationGln","representationIrreducible":"representationTheory.htm#representationIrreducible","representationTensorProduct":"representationTheory.htm#representationTensorProduct","representationTheory":"representationTheory.htm#representationTheory","researchStoryLinrec":"researchStories.htm#researchStoryLinrec","researchStoryTable":"researchStories.htm#researchStoryTable","resources":"resources.htm#resources","restrictedEulerianPolynomial":"realRootedWords.htm#restrictedEulerianPolynomial","restrictedPermutationDescentsRealRooted":"realRootedWords.htm#restrictedPermutationDescentsRealRooted","restrictedSchur":"restrictedSchurFunctions.htm#restrictedSchur","retStatistic":"shuffleAndDelta.htm#retStatistic","reverseComplementOperator":"tableauOperators.htm#reverseComplementOperator","reverseSemistandardYoungTableau":"schurShifted.htm#reverseSemistandardYoungTableau","ribbonDefinition":"preliminaries.htm#ribbonDefinition","ribbonHead":"preliminaries.htm#ribbonHead","ribbonTail":"preliminaries.htm#ribbonTail","richardsonVarieties":"varieties.htm#richardsonVarieties","richardsonVarietyDefinition":"varieties.htm#richardsonVarietyDefinition","riggedConfigurations":"riggedConfigurations.htm#riggedConfigurations","rightKey":"key.htm#rightKey","rigidityMatroid":"matroids.htm#rigidityMatroid","rimHook":"preliminaries.htm#rimHook","rimHookRSK":"rsk.htm#rimHookRSK","riordanArrayPF":"polyaFrequency.htm#riordanArrayPF","riseDeltaConjecture":"shuffleAndDelta.htm#riseDeltaConjecture","roichmanChi":"murnaghanNakayama.htm#roichmanChi","rookLatticePathMatroids":"lattice-path-matroids.htm#rookLatticePathMatroids","rookLatticePathPolytopes":"lattice-path-matroids.htm#rookLatticePathPolytopes","rookMatroidDefinition":"lattice-path-matroids.htm#rookMatroidDefinition","rookMatroidPolynomials":"lattice-path-matroids.htm#rookMatroidPolynomials","rookMatroids":"lattice-path-matroids.htm#rookMatroids","rookNesting":"lattice-path-matroids.htm#rookNesting","rookPolynomial":"realRootedGraphs.htm#rookPolynomial","rookPolynomials":"realRootedGraphs.htm#rookPolynomials","rookTheory":"chromaticQuasisymmetric.htm#rookTheory","rootIdeal":"kschur.htm#rootIdeal","rootOrder":"realRootedInterlacing.htm#rootOrder","rootPolytope":"polytopes.htm#rootPolytope","rootPolytopeDefinition":"polytopes.htm#rootPolytopeDefinition","rootSystem":"root-systems.htm#rootSystem","rootSystemClassification":"root-systems.htm#rootSystemClassification","rootedLabeledTreeDefinition":"parking-functions.htm#rootedLabeledTreeDefinition","rootedTreePoset":"posets.htm#rootedTreePoset","rotheDiagram":"permutations.htm#rotheDiagram","rotheDiagramDefinition":"permutations.htm#rotheDiagramDefinition","rowGeneratingFunction":"realRooted.htm#rowGeneratingFunction","rowInsertion":"rsk.htm#rowInsertion","rowStrictDualImmaculateSchurFunction":"qsymSchur.htm#rowStrictDualImmaculateSchurFunction","rowStrictExtendedSchurFunction":"qsymSchur.htm#rowStrictExtendedSchurFunction","rowStrictImmaculateTableau":"qsymSchur.htm#rowStrictImmaculateTableau","rowStrictQuasisymmetricSchurFunction":"qsymSchur.htm#rowStrictQuasisymmetricSchurFunction","rowStrictYoungQuasisymmetricSchurFunction":"qsymSchur.htm#rowStrictYoungQuasisymmetricSchurFunction","rowmotion":"rowmotionPromotion.htm#rowmotion","rowmotionPromotion":"rowmotionPromotion.htm#rowmotionPromotion","rowmotionPromotionConjugacy":"rowmotionPromotion.htm#rowmotionPromotionConjugacy","rowmotionTamari":"rowmotionPromotion.htm#rowmotionTamari","rsExtSchur":"qsymSchur.htm#rsExtSchur","rsdiSchur":"qsymSchur.htm#rsdiSchur","rsk":"rsk.htm#rsk","rskBurge":"rsk.htm#rskBurge","rskCorrespondence":"rsk.htm#rskCorrespondence","rskDual":"rsk.htm#rskDual","rskOne":"rsk.htm#rskOne","rskProperties":"rsk.htm#rskProperties","rskStandardization":"rsk.htm#rskStandardization","rskYangBaxter":"rsk.htm#rskYangBaxter","rsqSchur":"qsymSchur.htm#rsqSchur","rsyqSchur":"qsymSchur.htm#rsyqSchur","runSortedDescentPolynomial":"realRootedWords.htm#runSortedDescentPolynomial","sEulerianPolynomial":"realRootedWords.htm#sEulerianPolynomial","sPerpTrick":"plethysm.htm#sPerpTrick","samePhaseCompatible":"stablePolynomials.htm#samePhaseCompatible","samePhaseStabilityDefinition":"stablePolynomials.htm#samePhaseStabilityDefinition","samePhaseStable":"stablePolynomials.htm#samePhaseStable","schedule":"shuffleAndDelta.htm#schedule","schubert":"schubert.htm#schubert","schubertAffine":"schubertVariations.htm#schubertAffine","schubertAsSumOfElementary":"schubert.htm#schubertAsSumOfElementary","schubertBCD":"schubertVariations.htm#schubertBCD","schubertBackStable":"backStableSchubert.htm#schubertBackStable","schubertBackStableDouble":"schubertVariations.htm#schubertBackStableDouble","schubertBlockDecomposition":"schubert.htm#schubertBlockDecomposition","schubertBumplessPipeDream":"schubert.htm#schubertBumplessPipeDream","schubertCalculus":"schubert-calculus.htm#schubertCalculus","schubertCalculusComputationRules":"schubert-calculus.htm#schubertCalculusComputationRules","schubertCalculusDefinition":"schubert-calculus.htm#schubertCalculusDefinition","schubertCalculusFlagVariety":"schubert-calculus.htm#schubertCalculusFlagVariety","schubertCalculusGiambelli":"schubert-calculus.htm#schubertCalculusGiambelli","schubertCalculusGr24":"schubert-calculus.htm#schubertCalculusGr24","schubertCalculusLRTruncation":"schubert-calculus.htm#schubertCalculusLRTruncation","schubertCalculusPieri":"schubert-calculus.htm#schubertCalculusPieri","schubertCalculusPoincareDuality":"schubert-calculus.htm#schubertCalculusPoincareDuality","schubertCauchyFormula":"schubert.htm#schubertCauchyFormula","schubertCharge":"various-research.htm#schubertCharge","schubertComplexity":"schubert.htm#schubertComplexity","schubertDouble":"schubertVariations.htm#schubertDouble","schubertDoubleGiambelliFormula":"schubertVariations.htm#schubertDoubleGiambelliFormula","schubertDual":"schubertVariations.htm#schubertDual","schubertEnriched":"schubertVariations.htm#schubertEnriched","schubertGTFormula":"schubert.htm#schubertGTFormula","schubertInvolution":"schubertVariations.htm#schubertInvolution","schubertKeyExpansion":"schubert.htm#schubertKeyExpansion","schubertLittlewoodRichardson":"schubert.htm#schubertLittlewoodRichardson","schubertMatroidDefinition":"varieties.htm#schubertMatroidDefinition","schubertMonksRule":"schubert.htm#schubertMonksRule","schubertMurnaghanNakayama":"schubert.htm#schubertMurnaghanNakayama","schubertOperatorformula":"schubert.htm#schubertOperatorformula","schubertOrthodontia":"schubert.htm#schubertOrthodontia","schubertPieriRule":"schubert.htm#schubertPieriRule","schubertPipeDream":"schubert.htm#schubertPipeDream","schubertQuantum":"schubertVariations.htm#schubertQuantum","schubertQuantumDouble":"schubertVariations.htm#schubertQuantumDouble","schubertQuantumDoubleCauchyIdentity":"schubertVariations.htm#schubertQuantumDoubleCauchyIdentity","schubertQuantumMonksRule":"schubertVariations.htm#schubertQuantumMonksRule","schubertQuantumMurnaghanNakayama":"schubertVariations.htm#schubertQuantumMurnaghanNakayama","schubertQuantumPieriRule":"schubertVariations.htm#schubertQuantumPieriRule","schubertQuantumSottileRule":"schubertVariations.htm#schubertQuantumSottileRule","schubertReducedwordFormula":"schubert.htm#schubertReducedwordFormula","schubertRepresentationTheory":"schubert.htm#schubertRepresentationTheory","schubertSkew":"schubertVariations.htm#schubertSkew","schubertSottileRule":"schubert.htm#schubertSottileRule","schubertTwisted":"schubertVariations.htm#schubertTwisted","schubertUniversal":"schubertVariations.htm#schubertUniversal","schubertVariations":"schubertVariations.htm#schubertVariations","schubertVarietiesAndMatroids":"varieties.htm#schubertVarietiesAndMatroids","schubertVarietyDefinition":"schubert-calculus.htm#schubertVarietyDefinition","schubertVarietyGeometryDefinition":"varieties.htm#schubertVarietyGeometryDefinition","schurAffine":"stanleySymmetric.htm#schurAffine","schurAffinePieriOperators":"stanleySymmetric.htm#schurAffinePieriOperators","schurAffineSkew":"stanleySymmetric.htm#schurAffineSkew","schurBottom":"schurMisc.htm#schurBottom","schurCauchyFormula":"schur.htm#schurCauchyFormula","schurCombinatorialFormula":"schur.htm#schurCombinatorialFormula","schurCumulant":"schur.htm#schurCumulant","schurCumulants":"schur.htm#schurCumulants","schurCylindric":"cylindricSchur.htm#schurCylindric","schurEhrhartRealRooted":"realRootedTableaux.htm#schurEhrhartRealRooted","schurEisensteinSeriesDefinition":"schurZeta.htm#schurEisensteinSeriesDefinition","schurFactorial":"schurShifted.htm#schurFactorial","schurFactorialAlternant":"schurShifted.htm#schurFactorialAlternant","schurFactorialJT":"schurShifted.htm#schurFactorialJT","schurFactorialLR":"schurShifted.htm#schurFactorialLR","schurFactorialMolevSagan":"schurShifted.htm#schurFactorialMolevSagan","schurFlagged":"schurFlagged.htm#schurFlagged","schurFunction":"schur.htm#schurFunction","schurGeneratingFunction":"schur.htm#schurGeneratingFunction","schurGeneratingFunctions":"schur.htm#schurGeneratingFunctions","schurGiambelliFormula":"schur.htm#schurGiambelliFormula","schurGustafsonMilne":"schur.htm#schurGustafsonMilne","schurHallInnerProduct":"schur.htm#schurHallInnerProduct","schurHanlonFormula":"schur.htm#schurHanlonFormula","schurHookCauchy":"superSymmetricSchur.htm#schurHookCauchy","schurHookDeterminant":"superSymmetricSchur.htm#schurHookDeterminant","schurHookGeneralizations":"superSymmetricSchur.htm#schurHookGeneralizations","schurHookJacobiTrudi":"superSymmetricSchur.htm#schurHookJacobiTrudi","schurHookLittlewoodFormula":"superSymmetricSchur.htm#schurHookLittlewoodFormula","schurHookProperties":"superSymmetricSchur.htm#schurHookProperties","schurInNonCom":"ncSchur.htm#schurInNonCom","schurJacobiTrudi":"schur.htm#schurJacobiTrudi","schurKP":"schurKP.htm#schurKP","schurKPQ":"schurKP.htm#schurKPQ","schurKPQLRRule":"schurKP.htm#schurKPQLRRule","schurKPQProps":"schurKP.htm#schurKPQProps","schurKQ":"schurKP.htm#schurKQ","schurKawanaka":"schur.htm#schurKawanaka","schurKroneckerCoefficients":"schur.htm#schurKroneckerCoefficients","schurLectureHall":"schurMisc.htm#schurLectureHall","schurLittlewoodRichardson":"schur.htm#schurLittlewoodRichardson","schurLoop":"loopSchur.htm#schurLoop","schurLoopDefinition":"loopSchur.htm#schurLoopDefinition","schurLoopJacobiTrudi":"loopSchur.htm#schurLoopJacobiTrudi","schurLoopMurnaghanNakayama":"loopSchur.htm#schurLoopMurnaghanNakayama","schurLoopWeylFormula":"loopSchur.htm#schurLoopWeylFormula","schurMiscRootTwists":"schurSymplecticOrthogonal.htm#schurMiscRootTwists","schurMurnaghanNakayama":"schur.htm#schurMurnaghanNakayama","schurOrthogonal":"schurSymplecticOrthogonal.htm#schurOrthogonal","schurOrthogonalCauchy":"schurSymplecticOrthogonal.htm#schurOrthogonalCauchy","schurOrthogonalJacobiTrudi":"schurSymplecticOrthogonal.htm#schurOrthogonalJacobiTrudi","schurOrthogonalOtherProperties":"schurSymplecticOrthogonal.htm#schurOrthogonalOtherProperties","schurP":"schurPQ.htm#schurP","schurPEvacuation":"schurPQ.htm#schurPEvacuation","schurPFunction":"schurPQ.htm#schurPFunction","schurPLRRule":"schurPQ.htm#schurPLRRule","schurPSchurExpansion":"schurPQ.htm#schurPSchurExpansion","schurPieriRule":"schur.htm#schurPieriRule","schurPlethysmCoefficient":"plethysm.htm#schurPlethysmCoefficient","schurPositive":"various-research.htm#schurPositive","schurPositiveDefinition":"schurPositivity.htm#schurPositiveDefinition","schurPositivity":"schurPositivity.htm#schurPositivity","schurPositivityCrystalSkeletons":"schurPositivity.htm#schurPositivityCrystalSkeletons","schurPositivityCrystals":"schurPositivity.htm#schurPositivityCrystals","schurPositivityDualEquivalence":"schurPositivity.htm#schurPositivityDualEquivalence","schurPositivityEG":"schurPositivity.htm#schurPositivityEG","schurPositivityPermutationClasses":"schurPositivity.htm#schurPositivityPermutationClasses","schurPositivityRSK":"schurPositivity.htm#schurPositivityRSK","schurPositivityRep":"schurPositivity.htm#schurPositivityRep","schurPositivitySlinky":"schurPositivity.htm#schurPositivitySlinky","schurQ":"schurPQ.htm#schurQ","schurQCauchy":"schurPQ.htm#schurQCauchy","schurQFunction":"schurPQ.htm#schurQFunction","schurRaisingFormula":"schur.htm#schurRaisingFormula","schurRibbon":"schurRibbon.htm#schurRibbon","schurRibbonIntoH":"schurRibbon.htm#schurRibbonIntoH","schurRibbonProduct":"schurRibbon.htm#schurRibbonProduct","schurRoichmanFormula":"schur.htm#schurRoichmanFormula","schurS":"schur.htm#schurS","schurSS":"schurSS.htm#schurSS","schurSVarious":"schur.htm#schurSVarious","schurShifted":"schurShifted.htm#schurShifted","schurShiftedCombinatorialFormula":"schurShifted.htm#schurShiftedCombinatorialFormula","schurShiftedJacobiTrudi":"schurShifted.htm#schurShiftedJacobiTrudi","schurShiftedLittlewoodRichardson":"schurShifted.htm#schurShiftedLittlewoodRichardson","schurShiftedProperties":"schurShifted.htm#schurShiftedProperties","schurSkew":"schur.htm#schurSkew","schurSkewAffineInschurAffine":"stanleySymmetric.htm#schurSkewAffineInschurAffine","schurSpecializations":"schur.htm#schurSpecializations","schurStripDecompFormula":"schur.htm#schurStripDecompFormula","schurSums":"schur.htm#schurSums","schurSymplectic":"schurSymplecticOrthogonal.htm#schurSymplectic","schurSymplecticBialternant":"schurSymplecticOrthogonal.htm#schurSymplecticBialternant","schurSymplecticCauchy":"schurSymplecticOrthogonal.htm#schurSymplecticCauchy","schurSymplecticCrystal":"schurSymplecticOrthogonal.htm#schurSymplecticCrystal","schurSymplecticGiambelli":"schurSymplecticOrthogonal.htm#schurSymplecticGiambelli","schurSymplecticJacobiTrudi":"schurSymplecticOrthogonal.htm#schurSymplecticJacobiTrudi","schurSymplecticMurnaghanNakayama":"schurSymplecticOrthogonal.htm#schurSymplecticMurnaghanNakayama","schurSymplecticOtherProperties":"schurSymplecticOrthogonal.htm#schurSymplecticOtherProperties","schurSymplecticQ":"symplecticQ.htm#schurSymplecticQ","schurSymplecticQPieri":"symplecticQ.htm#schurSymplecticQPieri","schurSymplecticQTab":"symplecticQ.htm#schurSymplecticQTab","schurSzegoComposition":"realRooted.htm#schurSzegoComposition","schurSzegoCompositionDefinition":"realRooted.htm#schurSzegoCompositionDefinition","schurSzegoInterlacing":"realRootedInterlacing.htm#schurSzegoInterlacing","schurToric":"cylindricSchur.htm#schurToric","schurWeylGraph":"representationTheory.htm#schurWeylGraph","schurWeylformula":"schur.htm#schurWeylformula","schurWreath":"schurMisc.htm#schurWreath","schurZeta":"schurZeta.htm#schurZeta","schurZetaP":"schurZeta.htm#schurZetaP","schursPPfaffian":"schurPQ.htm#schursPPfaffian","schursPTableauformula":"schurPQ.htm#schursPTableauformula","semistandardAugmentedFilling":"key.htm#semistandardAugmentedFilling","semistandardReverseCompositionTableau":"qsymSchur.htm#semistandardReverseCompositionTableau","semistandardYoungTableau":"preliminaries.htm#semistandardYoungTableau","semistandardYoungTableauOfType":"preliminaries.htm#semistandardYoungTableauOfType","separableDescentRealRooted":"realRootedWords.htm#separableDescentRealRooted","sergeevPragaczFormula":"superSymmetricSchur.htm#sergeevPragaczFormula","seriesParallelPoset":"posets.htm#seriesParallelPoset","setPartitions":"combinatorialObjects.htm#setPartitions","setSystem":"matroids.htm#setSystem","setValuedSkylineFilling":"lascoux.htm#setValuedSkylineFilling","setValuedTableaux":"combinatorialObjects.htm#setValuedTableaux","shareshianWachsConjecture":"chromaticQuasisymmetric.htm#shareshianWachsConjecture","shellingInterlacingCriterion":"realRootedInterlacing.htm#shellingInterlacingCriterion","shiArrangementDefinition":"parking-functions.htm#shiArrangementDefinition","shiftedSchurLittlewoodRichardsonCoefficient":"schurShifted.htm#shiftedSchurLittlewoodRichardsonCoefficient","shiftedSchurPolynomialDefinition":"schurShifted.htm#shiftedSchurPolynomialDefinition","shiftedSchurVanishingProperty":"schurShifted.htm#shiftedSchurVanishingProperty","shiftedSupersymmetricSchurPolynomial":"superSymmetricSchur.htm#shiftedSupersymmetricSchurPolynomial","shiftedSymmetricFunctionDefinition":"jackShifted.htm#shiftedSymmetricFunctionDefinition","shiftedTableau":"schurPQ.htm#shiftedTableau","shuffleOfPermutations":"gessel.htm#shuffleOfPermutations","signedChromaticPolynomial":"chromaticQuasisymmetric.htm#signedChromaticPolynomial","signedChromaticQuasisymmetricFunction":"chromaticQuasisymmetric.htm#signedChromaticQuasisymmetricFunction","signedPermutationsRSK":"rsk.htm#signedPermutationsRSK","simpleTranspositionDefinition":"permutations.htm#simpleTranspositionDefinition","singleRowMacdonaldCumulantLLTPositivity":"macdonaldH.htm#singleRowMacdonaldCumulantLLTPositivity","skewAffineStanleySymmetricFunction":"stanleySymmetric.htm#skewAffineStanleySymmetricFunction","skewDiagramContent":"llt.htm#skewDiagramContent","skewDualStableGrothendieckPolynomial":"grothendieck.htm#skewDualStableGrothendieckPolynomial","skewFactorialSchurFunction":"schurShifted.htm#skewFactorialSchurFunction","skewGTEhrhartPositivity":"gtpatterns.htm#skewGTEhrhartPositivity","skewHallLittlewoodP":"hallLittlewood.htm#skewHallLittlewoodP","skewHiveDefinition":"hivePolytopes.htm#skewHiveDefinition","skewHives":"hivePolytopes.htm#skewHives","skewJacobiTrudi":"schur.htm#skewJacobiTrudi","skewKostkaFoulkes":"kostkaFoulkes.htm#skewKostkaFoulkes","skewKostkaFoulkesPolynomial":"kostkaFoulkes.htm#skewKostkaFoulkesPolynomial","skewQWhittakerFunction":"whittaker.htm#skewQWhittakerFunction","skewQuasisymmetricSchurFunctions":"qsymSchur.htm#skewQuasisymmetricSchurFunctions","skewRSK":"rsk.htm#skewRSK","skewSYT":"combinatorialObjects.htm#skewSYT","skewSchurAsSchubert":"schubert.htm#skewSchurAsSchubert","skewSchurCombinatorialFormula":"schur.htm#skewSchurCombinatorialFormula","skewSchurFiniteSchubertSpecialization":"schubert.htm#skewSchurFiniteSchubertSpecialization","skewSchurGesselExpansion":"schur.htm#skewSchurGesselExpansion","skewSchurLittlewood":"schur.htm#skewSchurLittlewood","skewSchurMurnaghanNakayama":"schur.htm#skewSchurMurnaghanNakayama","skewSchurNoncommuting":"ncSchur.htm#skewSchurNoncommuting","skewSchurPolynomial":"schur.htm#skewSchurPolynomial","skewSchurSpecializations":"schur.htm#skewSchurSpecializations","skewShapeDefinition":"preliminaries.htm#skewShapeDefinition","skewShapes":"preliminaries.htm#skewShapes","skewingOperator":"schur.htm#skewingOperator","skewqWhittaker":"whittaker.htm#skewqWhittaker","slideF":"assaf.htm#slideF","slideM":"assaf.htm#slideM","slidePositive":"assaf.htm#slidePositive","smallInterlacingMatrices":"realRootedInterlacing.htm#smallInterlacingMatrices","snakeMatroidShardPolytopes":"lattice-path-matroids.htm#snakeMatroidShardPolytopes","socleTableau":"littlewoodRichardson.htm#socleTableau","socleTableaux":"littlewoodRichardson.htm#socleTableaux","softwareLinks":"resources.htm#softwareLinks","sortedTableauxMacdonald":"macdonaldH.htm#sortedTableauxMacdonald","sourceSchur":"ncSchur.htm#sourceSchur","sourceSchurNoncommuting":"ncSchur.htm#sourceSchurNoncommuting","sourceSkewSchurNoncommuting":"ncSchur.htm#sourceSkewSchurNoncommuting","southwestDiagram":"kohnert.htm#southwestDiagram","spaceSym":"standardSymmetricFunctions.htm#spaceSym","sparsePavingMatroid":"matroids.htm#sparsePavingMatroid","spechtFrobenius":"specht-modules.htm#spechtFrobenius","spechtModuleClassicalDefinition":"specht-modules.htm#spechtModuleClassicalDefinition","spechtModuleDefinition":"stanleySymmetric.htm#spechtModuleDefinition","spechtModules":"specht-modules.htm#spechtModules","spechtModulesIntro":"specht-modules.htm#spechtModulesIntro","spechtRecentWork":"specht-modules.htm#spechtRecentWork","spechtTabloids":"specht-modules.htm#spechtTabloids","spechtZeroHeckeComparison":"specht-modules.htm#spechtZeroHeckeComparison","specialOrthogonalTableau":"schurSymplecticOrthogonal.htm#specialOrthogonalTableau","specialRimHookTabloid":"standardSymmetricFunctions.htm#specialRimHookTabloid","specializedDescentPolynomialsRealRooted":"realRootedWords.htm#specializedDescentPolynomialsRealRooted","spinCharacter":"schurPQ.htm#spinCharacter","spinHallLittlewoodRationalFunction":"hallLittlewood.htm#spinHallLittlewoodRationalFunction","spinLLTPolynomial":"llt.htm#spinLLTPolynomial","springerFiberDefinition":"varieties.htm#springerFiberDefinition","springerFibers":"varieties.htm#springerFibers","squarePathLLTSchedules":"shuffleAndDelta.htm#squarePathLLTSchedules","squarePathPreferenceFunctions":"shuffleAndDelta.htm#squarePathPreferenceFunctions","squarePathSpecializations":"shuffleAndDelta.htm#squarePathSpecializations","squarePathTheorem":"shuffleAndDelta.htm#squarePathTheorem","ssytAsLatticePaths":"preliminaries.htm#ssytAsLatticePaths","stabilityConnections":"stablePolynomials.htm#stabilityConnections","stabilityOperators":"stablePolynomials.htm#stabilityOperators","stabilitySymbol":"stablePolynomials.htm#stabilitySymbol","stabilitymultiaffine":"stablePolynomials.htm#stabilitymultiaffine","stableCapacity":"stablePolynomials.htm#stableCapacity","stableGrothendieckPolynomial":"grothendieck.htm#stableGrothendieckPolynomial","stableMatchingNegativeDependence":"stablePolynomials.htm#stableMatchingNegativeDependence","stablePolynomial":"stablePolynomials.htm#stablePolynomial","stablePolynomialDefinition":"stablePolynomials.htm#stablePolynomialDefinition","stablePrincipalSpecialization":"standardSymmetricFunctions.htm#stablePrincipalSpecialization","stableSpinHallLittlewoodFunction":"hallLittlewood.htm#stableSpinHallLittlewoodFunction","staircaseSchurRoots":"various-research.htm#staircaseSchurRoots","standardBasis":"key.htm#standardBasis","standardElementaryMonomial":"backStableSchubert.htm#standardElementaryMonomial","standardIdentities":"standardSymmetricFunctions.htm#standardIdentities","standardImmaculateTableau":"qsymSchur.htm#standardImmaculateTableau","standardInvolution":"symmetricFunctionsSpace.htm#standardInvolution","standardNewton":"standardSymmetricFunctions.htm#standardNewton","standardPeakCompositionTableau":"qsymSchur.htm#standardPeakCompositionTableau","standardPeakYoungCompositionTableau":"qsymSchur.htm#standardPeakYoungCompositionTableau","standardQuasiInvolution":"standardQuasiSymmetricFunctions.htm#standardQuasiInvolution","standardReverseCompositionTableau":"qsymSchur.htm#standardReverseCompositionTableau","standardSchurNoncommuting":"ncSchur.htm#standardSchurNoncommuting","standardSpecialization":"standardSymmetricFunctions.htm#standardSpecialization","standardSubword":"kostkaFoulkes.htm#standardSubword","standardSubwords":"kostkaFoulkes.htm#standardSubwords","standardYoungTableau":"preliminaries.htm#standardYoungTableau","standardizedReadingWordPreferenceFunction":"parking-functions.htm#standardizedReadingWordPreferenceFunction","stanleyDecreasingFactorizationDefinition":"stanleySymmetric.htm#stanleyDecreasingFactorizationDefinition","stanleyMonotonicity":"polytopes.htm#stanleyMonotonicity","stanleySchurPositiveRealRooted":"polyaFrequency.htm#stanleySchurPositiveRealRooted","stanleyStembridgeConjecture":"chromaticQuasisymmetric.htm#stanleyStembridgeConjecture","stanleySym":"stanleySymmetric.htm#stanleySym","stanleySymAffine":"stanleySymmetric.htm#stanleySymAffine","stanleySymAffineCoproduct":"stanleySymmetric.htm#stanleySymAffineCoproduct","stanleySymAffineFPF":"stanleySymmetric.htm#stanleySymAffineFPF","stanleySymAffineInvolution":"stanleySymmetric.htm#stanleySymAffineInvolution","stanleySymAffineSchurAffineExpansion":"stanleySymmetric.htm#stanleySymAffineSchurAffineExpansion","stanleySymAffineSchurExpansion":"stanleySymmetric.htm#stanleySymAffineSchurExpansion","stanleySymAffineSkew":"stanleySymmetric.htm#stanleySymAffineSkew","stanleySymBC":"stanleySymmetric.htm#stanleySymBC","stanleySymBCSchurExpansion":"stanleySymmetric.htm#stanleySymBCSchurExpansion","stanleySymDouble":"stanleySymmetric.htm#stanleySymDouble","stanleySymFrobenius":"stanleySymmetric.htm#stanleySymFrobenius","stanleySymInvolution":"stanleySymmetric.htm#stanleySymInvolution","stanleySymLimitDefinition":"stanleySymmetric.htm#stanleySymLimitDefinition","stanleySymReducedWordDefinition":"stanleySymmetric.htm#stanleySymReducedWordDefinition","stanleySymSchurExpansion":"stanleySymmetric.htm#stanleySymSchurExpansion","stanleySymmetricFunction":"stanleySymmetric.htm#stanleySymmetricFunction","stasheffPolytope":"polytopes.htm#stasheffPolytope","steinbergFormula":"kostkaFoulkes.htm#steinbergFormula","steinbergLRFormula":"littlewoodRichardson.htm#steinbergLRFormula","stembridgeCrystal":"crystals.htm#stembridgeCrystal","stirlingPermutationDescentPolynomial":"realRootedSetPartitions.htm#stirlingPermutationDescentPolynomial","stretchedLRPolynomial":"hivePolytopes.htm#stretchedLRPolynomial","strictGTPattern":"gtpatterns.htm#strictGTPattern","strictHurwitzStablePolynomial":"realRootedInterlacing.htm#strictHurwitzStablePolynomial","strictPartition":"schurPQ.htm#strictPartition","strongDeltaMatroid":"matroids.htm#strongDeltaMatroid","strongOrderDefinition":"permutations.htm#strongOrderDefinition","stronglyNiceProperty":"chromaticQuasisymmetric.htm#stronglyNiceProperty","stronglyRayleigh":"stablePolynomials.htm#stronglyRayleigh","stronglyRayleighProperty":"stablePolynomials.htm#stronglyRayleighProperty","sturmEulerian":"realRootedWords.htm#sturmEulerian","sturmOther":"realRootedSetPartitions.htm#sturmOther","sturmPermutations":"realRootedWords.htm#sturmPermutations","sturmRootCounting":"realRootedInterlacing.htm#sturmRootCounting","sturmSequence":"realRootedInterlacing.htm#sturmSequence","sturmSequenceDefinition":"realRootedInterlacing.htm#sturmSequenceDefinition","sturmSetPartitions":"realRootedSetPartitions.htm#sturmSetPartitions","sturmUnimodality":"realRootedInterlacing.htm#sturmUnimodality","submodular":"matroids.htm#submodular","submodularFunction":"matroids.htm#submodularFunction","subsetCyclicSieving":"cyclic-sieving.htm#subsetCyclicSieving","subspaceProfiles":"whittaker.htm#subspaceProfiles","substringCoalgebra":"standardQuasiSymmetricFunctions.htm#substringCoalgebra","sulzgruberInsertion":"rsk.htm#sulzgruberInsertion","superJackPolynomial":"jack.htm#superJackPolynomial","superNablaOperator":"diagonalHarmonics.htm#superNablaOperator","superSymmetricFunctions":"superSymmetricSchur.htm#superSymmetricFunctions","supersymmetricFunction":"superSymmetricSchur.htm#supersymmetricFunction","supersymmetricLLTPolynomial":"llt.htm#supersymmetricLLTPolynomial","supersymmetricMonomialPolynomial":"superSymmetricSchur.htm#supersymmetricMonomialPolynomial","supersymmetricPowerSumPolynomial":"superSymmetricSchur.htm#supersymmetricPowerSumPolynomial","sxpRule":"plethysm.htm#sxpRule","sylvesterSimplex":"polytopes.htm#sylvesterSimplex","symmetricDecompInterlacing":"realRootedInterlacing.htm#symmetricDecompInterlacing","symmetricDecomposition":"gammaPositivity.htm#symmetricDecomposition","symmetricDecompositionMethod":"gammaPositivity.htm#symmetricDecompositionMethod","symmetricEdgePolytope":"polytopes.htm#symmetricEdgePolytope","symmetricExchangeBinomial":"matroids.htm#symmetricExchangeBinomial","symmetricFunction":"standardSymmetricFunctions.htm#symmetricFunction","symmetricFunctionAntipode":"symmetricFunctionsSpace.htm#symmetricFunctionAntipode","symmetricFunctions":"standardSymmetricFunctions.htm#symmetricFunctions","symmetricFunctionsSpace":"symmetricFunctionsSpace.htm#symmetricFunctionsSpace","symmetricGeneratingFamilies":"symmetricFunctionsSpace.htm#symmetricGeneratingFamilies","symmetricIDecomposition":"realRootedInterlacing.htm#symmetricIDecomposition","symmetricPolynomial":"gammaPositivity.htm#symmetricPolynomial","symmetricTransitionMatrices":"standardSymmetricFunctions.htm#symmetricTransitionMatrices","symplecticKostkaFoulkes":"kostkaFoulkes.htm#symplecticKostkaFoulkes","symplecticRajchgotIndex":"grothendieck.htm#symplecticRajchgotIndex","symplecticTableau":"schurSymplecticOrthogonal.htm#symplecticTableau","sytDescentRealRooted":"realRootedTableaux.htm#sytDescentRealRooted","tAtom":"macdonaldEperm.htm#tAtom","tableauDefinition":"preliminaries.htm#tableauDefinition","tableauDescentSet":"preliminaries.htm#tableauDescentSet","tableauEvacuation":"tableauOperators.htm#tableauEvacuation","tableauMajorIndex":"preliminaries.htm#tableauMajorIndex","tableauOpOther":"tableauOperators.htm#tableauOpOther","tableauOpRelations":"tableauOperators.htm#tableauOpRelations","tableauOperations":"tableauOperators.htm#tableauOperations","tableauPromotion":"tableauOperators.htm#tableauPromotion","tableauRowPermutation":"ncSchur.htm#tableauRowPermutation","tabloidDefinition":"specht-modules.htm#tabloidDefinition","tamariInterval":"shuffleAndDelta.htm#tamariInterval","temperleyLiebCrystal":"crystals.htm#temperleyLiebCrystal","tensorProductRepresentationDefinition":"representationTheory.htm#tensorProductRepresentationDefinition","teslerMatrix":"shuffleAndDelta.htm#teslerMatrix","thagomizerKazhdanLusztigRealRooted":"matroids.htm#thagomizerKazhdanLusztigRealRooted","thagomizerMatroid":"matroids.htm#thagomizerMatroid","thm:brenti":"polyaFrequency.htm#thm:brenti","thm:brentiFallingFactorialBasis":"realRooted.htm#thm:brentiFallingFactorialBasis","thm:keyGT":"key.htm#thm:keyGT","thm:narayanaTransformationMaoWang":"realRooted.htm#thm:narayanaTransformationMaoWang","thomPolynomial":"thomPolynomials.htm#thomPolynomial","thomPolynomials":"thomPolynomials.htm#thomPolynomials","threePlusOneFreePoset":"posets.htm#threePlusOneFreePoset","tnnLowerUnitriangularMatrix":"polyaFrequency.htm#tnnLowerUnitriangularMatrix","tnnMixingInterlacing":"realRootedInterlacing.htm#tnnMixingInterlacing","tnnPathMatrix":"polyaFrequency.htm#tnnPathMatrix","toeplitzTnnMatrix":"polyaFrequency.htm#toeplitzTnnMatrix","toricGContributionPolynomial":"polytopes.htm#toricGContributionPolynomial","toricPromotion":"tableauOperators.htm#toricPromotion","toricPromotionRefraction":"cyclic-sieving.htm#toricPromotionRefraction","toricSchurIntoSchur":"cylindricSchur.htm#toricSchurIntoSchur","toricSchurPolynomial":"cylindricSchur.htm#toricSchurPolynomial","toricShape":"cylindricSchur.htm#toricShape","toricSpechtModule":"cylindricSchur.htm#toricSpechtModule","totalChromaticQuasisymmetricFunction":"chromaticQuasisymmetric.htm#totalChromaticQuasisymmetricFunction","totalPositivity":"totalPositivity.htm#totalPositivity","totalPositivityGrassmannians":"totalPositivity.htm#totalPositivityGrassmannians","totalPositivityNetworks":"totalPositivity.htm#totalPositivityNetworks","totalPositivityPF":"totalPositivity.htm#totalPositivityPF","totalPositivityRealRootedness":"totalPositivity.htm#totalPositivityRealRootedness","totallyNonnegative":"polyaFrequency.htm#totallyNonnegative","totallyNonnegativeMatrix":"polyaFrequency.htm#totallyNonnegativeMatrix","totallyPositiveMatrix":"polyaFrequency.htm#totallyPositiveMatrix","touchardAsMatching":"realRootedGraphs.htm#touchardAsMatching","touchardPolynomial":"realRootedSetPartitions.htm#touchardPolynomial","touchardPolynomialDefinition":"realRootedSetPartitions.htm#touchardPolynomialDefinition","touchardRiordanAreaSequence":"touchardRiordan.htm#touchardRiordanAreaSequence","touchardRiordanPolynomialDefinition":"touchardRiordan.htm#touchardRiordanPolynomialDefinition","transcendentalStabilitySymbols":"stablePolynomials.htm#transcendentalStabilitySymbols","transcendentalSymbol":"stablePolynomials.htm#transcendentalSymbol","transferMatrixChoosingStates":"transfer-matrix.htm#transferMatrixChoosingStates","transferMatrixDominoTilings":"transfer-matrix.htm#transferMatrixDominoTilings","transferMatrixFibonacciWords":"transfer-matrix.htm#transferMatrixFibonacciWords","transferMatrixGraphColorings":"transfer-matrix.htm#transferMatrixGraphColorings","transferMatrixMethod":"transfer-matrix.htm#transferMatrixMethod","transferMatrixTriominoTilings":"transfer-matrix.htm#transferMatrixTriominoTilings","transferMatrixWeightedTilings":"transfer-matrix.htm#transferMatrixWeightedTilings","transformedHallLittlewoodPolynomial":"hallLittlewood.htm#transformedHallLittlewoodPolynomial","transversal":"matroids.htm#transversal","transversalMatroids":"matroids.htm#transversalMatroids","treesChromaticQuasisymmetric":"chromaticQuasisymmetric.htm#treesChromaticQuasisymmetric","truncatedKrylovSubspace":"whittaker.htm#truncatedKrylovSubspace","tutteInternalExternal":"matroids.htm#tutteInternalExternal","tuttePolynomial":"matroids.htm#tuttePolynomial","tuttePolynomialDefinition":"matroids.htm#tuttePolynomialDefinition","tutteProperties":"matroids.htm#tutteProperties","tutteSpanningTrees":"tutteSymmetric.htm#tutteSpanningTrees","tutteSymmetric":"tutteSymmetric.htm#tutteSymmetric","twistedSchubertPolynomial":"schubertVariations.htm#twistedSchubertPolynomial","twoPlusTwoFreePoset":"posets.htm#twoPlusTwoFreePoset","tymoczkoDotAction":"dot-action.htm#tymoczkoDotAction","typeATriple":"macdonaldEperm.htm#typeATriple","typeBCrystal":"crystals.htm#typeBCrystal","typeBEulerianPolynomial":"realRootedWords.htm#typeBEulerianPolynomial","typeBStanleySymmetricFunction":"stanleySymmetric.htm#typeBStanleySymmetricFunction","typeBTriple":"macdonaldEperm.htm#typeBTriple","typeCStanleySymmetricFunction":"stanleySymmetric.htm#typeCStanleySymmetricFunction","typeCUniversalCharacter":"hallLittlewood.htm#typeCUniversalCharacter","typeDEulerianPolynomial":"realRootedWords.htm#typeDEulerianPolynomial","typeTranslation":"coxeterGroups.htm#typeTranslation","unicellularLLT":"unicellular-llt.htm#unicellularLLT","unicellularLLTAscent":"unicellular-llt.htm#unicellularLLTAscent","unicellularLLTChromaticRelation":"unicellular-llt.htm#unicellularLLTChromaticRelation","unicellularLLTEExpansion":"unicellular-llt.htm#unicellularLLTEExpansion","unicellularLLTFrobeniusTarget":"unicellular-llt.htm#unicellularLLTFrobeniusTarget","unicellularLLTGraphModel":"unicellular-llt.htm#unicellularLLTGraphModel","unicellularLLTGraphPolynomial":"unicellular-llt.htm#unicellularLLTGraphPolynomial","unicellularLLTNonCom":"unicellular-llt.htm#unicellularLLTNonCom","unicellularLLTPExpansion":"unicellular-llt.htm#unicellularLLTPExpansion","unicellularLLTPolynomialDefinition":"unicellular-llt.htm#unicellularLLTPolynomialDefinition","unicellularLLTProperties":"unicellular-llt.htm#unicellularLLTProperties","unicellularLLTRecursions":"unicellular-llt.htm#unicellularLLTRecursions","unicellularLLTRepresentationTheory":"unicellular-llt.htm#unicellularLLTRepresentationTheory","unicellularLLTSchurExpansion":"unicellular-llt.htm#unicellularLLTSchurExpansion","unicellularLLTTwinGKM":"unicellular-llt.htm#unicellularLLTTwinGKM","uniformMatroid":"matroids.htm#uniformMatroid","unimodalFactorization":"stanleySymmetric.htm#unimodalFactorization","unimodalPolynomial":"realRooted.htm#unimodalPolynomial","unitIntervalAO":"touchardRiordan.htm#unitIntervalAO","unitIntervalCographsA152947":"various-research.htm#unitIntervalCographsA152947","unitIntervalGraph":"chromaticQuasisymmetric.htm#unitIntervalGraph","unitIntervalOrder":"posets.htm#unitIntervalOrder","uphoPoset":"polyaFrequency.htm#uphoPoset","upperHookLength":"jack.htm#upperHookLength","valleyDeltaConjecture":"shuffleAndDelta.htm#valleyDeltaConjecture","valleyHoppingAction":"gammaPositivity.htm#valleyHoppingAction","valuatedMatroid":"matroids.htm#valuatedMatroid","vamosMatroid":"matroids.htm#vamosMatroid","variationDiminishing":"polyaFrequency.htm#variationDiminishing","variationDiminishingProperty":"polyaFrequency.htm#variationDiminishingProperty","varieties":"varieties.htm#varieties","varietiesForKeys":"varieties.htm#varietiesForKeys","varietiesFurtherReading":"varieties.htm#varietiesFurtherReading","veroneseNumeratorOperator":"realRootedInterlacing.htm#veroneseNumeratorOperator","veroneseSectionsPF":"polyaFrequency.htm#veroneseSectionsPF","veroneseSectionsRealRooted":"polyaFrequency.htm#veroneseSectionsRealRooted","verschiebungOperator":"plethysm.htm#verschiebungOperator","verschiebungOperatorSymmetricFunctions":"symmetricFunctionsSpace.htm#verschiebungOperatorSymmetricFunctions","vexillary":"permutationFamilies.htm#vexillary","wagnerLemma":"realRootedInterlacing.htm#wagnerLemma","wangYehAffineCriterion":"realRootedInterlacing.htm#wangYehAffineCriterion","wangYehPfPreserver":"polyaFrequency.htm#wangYehPfPreserver","weakCompositionDefinition":"preliminaries.htm#weakCompositionDefinition","weakCompositions":"preliminaries.htm#weakCompositions","weakHurwitzStablePolynomial":"realRootedInterlacing.htm#weakHurwitzStablePolynomial","weakOrder":"permutations.htm#weakOrder","weakOrderDefinition":"permutations.htm#weakOrderDefinition","weakRayleighProperty":"stablePolynomials.htm#weakRayleighProperty","weightLifting":"polytopes.htm#weightLifting","weightRestrictedGTPolytope":"gtpatterns.htm#weightRestrictedGTPolytope","weightedBondChromaticSymmetric":"weightedBondSymmetric.htm#weightedBondChromaticSymmetric","weightedBondPoset":"weightedBondSymmetric.htm#weightedBondPoset","weightedBondSymmetric":"weightedBondSymmetric.htm#weightedBondSymmetric","weightedIndependencePolynomial":"realRootedGraphs.htm#weightedIndependencePolynomial","weylFormula":"schur.htm#weylFormula","weylLieGroups":"root-systems.htm#weylLieGroups","whitehouseModuleDefinition":"cycleIndexPolynomial.htm#whitehouseModuleDefinition","whitneyNumbersSecondKind":"matroids.htm#whitneyNumbersSecondKind","whittaker":"whittaker.htm#whittaker","wittSymmetric":"standardSymmetricFunctions.htm#wittSymmetric","wordList":"dictionary.htm#wordList","wordParkingFunctionDefinition":"parking-functions.htm#wordParkingFunctionDefinition","wordPreferenceFunctionDefinition":"parking-functions.htm#wordPreferenceFunctionDefinition","wordQuasiSymmetricFunctions":"standardQuasiSymmetricFunctions.htm#wordQuasiSymmetricFunctions","wordWeight":"preliminaries.htm#wordWeight","wreathInterpolationMacdonaldPolynomial":"macdonaldP.htm#wreathInterpolationMacdonaldPolynomial","wreathMacdonaldPolynomial":"macdonaldP.htm#wreathMacdonaldPolynomial","wreathProductSchurPolynomial":"schurMisc.htm#wreathProductSchurPolynomial","wronskianCriterion":"realRootedInterlacing.htm#wronskianCriterion","yamanouchiInequalities":"gtpatterns.htm#yamanouchiInequalities","yamanouchiWord":"preliminaries.htm#yamanouchiWord","yangBaxter":"yang-baxter.htm#yangBaxter","youngDiagram":"preliminaries.htm#youngDiagram","youngQuasisymmetricSchurFunction":"qsymSchur.htm#youngQuasisymmetricSchurFunction","youngsConstruction":"murnaghanNakayama.htm#youngsConstruction","yqSchur":"qsymSchur.htm#yqSchur","zeroHeckeAlgebra":"zeroHecke.htm#zeroHeckeAlgebra","zeroHeckeAlgebraDefinition":"zeroHecke.htm#zeroHeckeAlgebraDefinition","zeroHeckeOtherTypes":"zeroHecke.htm#zeroHeckeOtherTypes","zeroHeckeQSymNSym":"zeroHecke.htm#zeroHeckeQSymNSym","zeroHeckeRepresentationTheory":"zeroHecke.htm#zeroHeckeRepresentationTheory","zeroHeckeSchurLike":"zeroHecke.htm#zeroHeckeSchurLike","zonal":"zonal.htm#zonal","zonalPowerSumExpansion":"zonal.htm#zonalPowerSumExpansion"}