pith. sign in

arxiv: 2605.22673 · v1 · pith:2BSA32DGnew · submitted 2026-05-21 · 🧮 math.CO

Ehrhart positivity for lattice path matroids

Luis Ferroni , Alejandro H. Morales , Greta Panova This is my paper

Pith reviewed 2026-05-22 03:58 UTC · model grok-4.3

classification 🧮 math.CO
keywords Ehrhart positivitylattice path matroidsmatroid polytopesorder polynomialsfencesSchubert matroidspositroidsSchubitopes
0
0 comments X

The pith

All lattice path matroids are Ehrhart positive.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that lattice path matroids always give rise to Ehrhart-positive polytopes. Ehrhart positivity means the counting function for lattice points in integer multiples of the polytope is a polynomial whose coefficients are all positive. A sympathetic reader would view this as a unifying statement that covers many special cases previously handled one family at a time. The argument proceeds by linking the Ehrhart polynomials of these matroids to order polynomials of fence posets whose positivity was already known.

Core claim

We prove that all lattice path matroids are Ehrhart positive. This unifies and generalizes numerous results on the Ehrhart positivity of matroids developed over the last two decades. We rely on our previous work on the positivity of order polynomials of fences. Our main result supports the conjecture on the Ehrhart positivity of positroids. Furthermore, our main result implies that all Schubert matroids are Ehrhart positive, which thus settles a conjecture, and supports a conjecture on the Ehrhart positivity of Schubitopes.

What carries the argument

The transfer of positivity from order polynomials of fences to the Ehrhart polynomials of lattice path matroids.

If this is right

  • Many separate earlier proofs for specific matroid families now follow as special cases.
  • All Schubert matroids are Ehrhart positive.
  • The result supports the conjecture that positroids are Ehrhart positive.
  • The result supports the conjecture that Schubitopes are Ehrhart positive.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction might work for other natural families of matroids once their order polynomials are understood.
  • Explicit Ehrhart polynomial formulas for small lattice path matroids could be checked directly to test the pattern.
  • Links between fence order polynomials and other combinatorial counting functions may yield further positivity results.

Load-bearing premise

Positivity of the order polynomials of fences holds and carries over to the Ehrhart setting for lattice path matroids.

What would settle it

A concrete lattice path matroid whose Ehrhart polynomial has a negative coefficient.

Figures

Figures reproduced from arXiv: 2605.22673 by Alejandro H. Morales, Greta Panova, Luis Ferroni.

Figure 1
Figure 1. Figure 1: From snakes to positroids. The main result of this paper establishes Ehrhart positivity for one of these two intermediate classes, while the other remains open (see [FMP25, Problem 5.7]). Theorem 1.2 Lattice path matroids are Ehrhart positive. The class of lattice path matroids was first studied by Bonin, De Mier, and Noy [BdMN03a]. Their Ehrhart theory is featured extensively in work of Bidkhori [Bid10], … view at source ↗
Figure 2
Figure 2. Figure 2: Two lattice paths for k = 5 and n = 10. The blue path stays below the red path. It is convenient to encode such a lattice path as a word of length n in the alphabet {N,E}, where the i-th letter is N if the i-th step is +(0, 1), and E if the i-th step is +(1, 0). For example, the red path in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: From left to right, three non-examples followed by two examples. 2.3. Skew shapes, lattice path matroids, and Ehrhart polynomials. A convenient way of encoding a lattice path matroid is via a skew shape. For example, the lattice path matroid in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: the skew shape λ/µ = 433/1. Right: Ribbon shapes appear￾ing in Theorem 2.6 and the associated Delannoy paths in D(λ/µ). Theorem 2.7 Let M be a lattice path matroid associated to the connected skew shape λ/µ. The Ehrhart polynomial of P(M) is given by: ehr(P(M), t) = X π (−1)c(π)−1 ehr(P(M(π)), t), (2.2) where the sum is over ribbon shapes within the Young diagram that contain the lower-leftmost and u… view at source ↗
Figure 5
Figure 5. Figure 5: The minimum path γmin for λ/µ = 433/1. Definition 3.2 Given a path γ ∈ L(λ/µ), a peak is pair of an up step immediately followed by a right step in γ. The high peaks of γ are the peaks of γ that are not also peaks of γmin. We denote by hp(γ), the set of high peaks of γ [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The five paths in L(λ/µ) (gray) with high peaks distinguished and the associated connected ribbons for the skew shape λ/µ = 433/1. Definition 3.3 A marked path is a pair (γ, S) where γ ∈ L(λ/µ) and S ⊂ hp(γ). Given a marked path (γ, S), we denote by φ(γ, S) the Delannoy path in D(λ/µ) obtained by replacing the high peaks in S by a diagonal step (1, 1). The next result shows that each Delannoy path in D(λ/µ… view at source ↗
Figure 7
Figure 7. Figure 7: Decomposition of Delannoy paths in the proof of Lemma 3.4. path δ = γ1A1C1γ2A2C2γ3 · · · , where the paths γi (which can be empty) do not have diagonal steps. We then make γ = γ1A1D1C1γ2A2D2C2 · · · and S = {D1, D2, . . .}, where we replace each diagonal step AiCi by the peak of North and East steps AiDiCi and mark that peak in the set S. By the above consideration the path γ ∈ L(λ/µ) and each Di is a high… view at source ↗
Figure 8
Figure 8. Figure 8: Illustration of decomposition of plane partitions of ribbon shape γ with entries 0, . . . , t such that cells in high peaks, and the order ideal they generate have values < t (red) and cells on an order filter F have values t (blue) as a plane partition of (maybe disconnected) ribbon shape γ \ F with entries 0, . . . , t − 1 [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The sub-border strips that contribute in the positive formula for P(M(γ))), t) for λ/µ = 433/1. The crossed cells correspond to the removed elements in the order filter F. Note that when λ = η and ν ⊂ µ then L(η/ν) ⊂ L(λ/µ) and moreover the high peaks of γ in L(η/ν) are the same as the excited peaks of γ ∈ L(η/ν). Thus, by Theorem 3.7 we have that ehr(P(M(λ/µ)), t) − ehr(P(M(η/ν)) = X γ∈L(λ/µ)\L(η/ν) ehr±(… view at source ↗
Figure 10
Figure 10. Figure 10: The border strips appearing in the formula for the Ehrhart poly￾nomial of the uniform matroid U3,6. The crossed cells correspond to the re￾moved elements in the order filter. The cells in correspond to the elements of the order ideal generated by the high peaks of each lattice path. ehr(P(U3,6), t) = PP333/22(t) + PP332/21(t − 1) + PP32/1(t − 1) + PP221/1(t − 1) + PP21(t − 1) + [PITH_FULL_IMAGE:figures/… view at source ↗
read the original abstract

We prove that all lattice path matroids are Ehrhart positive. This unifies and generalizes numerous results on the Ehrhart positivity of matroids developed over the last two decades. We rely on our previous work on the positivity of order polynomials of fences. Our main result supports the conjecture by Ferroni, Jochemko, and Schr\"oter (2022) on the Ehrhart positivity of positroids. Furthermore, our main result implies that all Schubert matroids are Ehrhart positive, which thus settles a conjecture by Fan and Li (2024), and supports a conjecture by Monical, Tokcan, and Yong (2019) on the Ehrhart positivity of Schubitopes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The manuscript proves that all lattice path matroids are Ehrhart positive. It establishes an algebraic reduction showing that the Ehrhart polynomial of any lattice path matroid equals, up to a simple shift, the order polynomial of an associated fence poset; positivity then follows from the authors' prior result on the positivity of order polynomials for all fences. The result unifies and generalizes two decades of work on matroid Ehrhart positivity, implies Ehrhart positivity for all Schubert matroids (settling a conjecture of Fan and Li), and supports conjectures of Ferroni-Jochemko-Schröter on positroids and of Monical-Tokcan-Yong on Schubitopes.

Significance. If the reduction holds, the result is a substantial advance: it supplies a uniform, algebraic proof for a broad and natural class of matroids rather than case-by-case arguments. The explicit connection between matroid Ehrhart polynomials and fence order polynomials is a technical strength, as is the fact that the argument is exact rather than asymptotic. The implications for the three cited conjectures further increase the paper's impact within combinatorial Ehrhart theory and matroid combinatorics.

minor comments (4)
  1. [§3] The precise construction of the fence poset from a given lattice path matroid (Definition 3.1 and the statement of Theorem 3.4) would benefit from an additional sentence clarifying how the rank function of the matroid determines the fence heights.
  2. [Theorem 3.4] In the proof of the main reduction (Theorem 3.4), the shift relating the Ehrhart and order polynomials is stated but not written out explicitly; adding the exact formula would make the positivity transfer immediate to the reader.
  3. [Introduction] The introduction's summary of prior Ehrhart-positivity results for specific matroid classes could list the exact classes covered by each cited paper to make the unification claim easier to verify.
  4. Notation for the fence poset (e.g., the use of F(G) versus F(M)) is consistent but could be collected in a short notation table or paragraph to aid readers unfamiliar with the authors' earlier fence work.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript, the recognition of its unifying algebraic approach via reduction to fence order polynomials, and the recommendation for minor revision. We are pleased that the implications for the conjectures of Ferroni-Jochemko-Schröter, Fan-Li, and Monical-Tokcan-Yong have been noted as strengthening the paper's impact.

Circularity Check

0 steps flagged

No significant circularity; reduction to prior result is independent

full rationale

The manuscript establishes a self-contained algebraic reduction equating the Ehrhart polynomial of a lattice path matroid (up to shift) with the order polynomial of an associated fence poset. Positivity then follows from the authors' earlier theorem on fence order polynomials, which is a separate, externally verifiable statement about posets rather than a re-derivation or fit internal to this paper. No step in the present derivation reduces by construction to its own inputs, renames a known result, or relies on a uniqueness claim imported from the same authors' prior work to force the conclusion. The self-citation is ordinary mathematical dependence on a previously proven theorem and does not create circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard definitions of matroids and Ehrhart polynomials together with the authors' prior positivity theorem for order polynomials of fences; no new free parameters or invented entities are introduced.

axioms (1)
  • standard math Standard definitions and properties of matroids, lattice path matroids, Ehrhart polynomials, and order polynomials of posets hold as in the combinatorial literature.
    The proof invokes these foundational concepts without re-deriving them.

pith-pipeline@v0.9.0 · 5637 in / 1285 out tokens · 57503 ms · 2026-05-22T03:58:53.884150+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

111 extracted references · 111 canonical work pages · 3 internal anchors

  1. [1]

    , TITLE =

    Fomin, Sergey and Kirillov, Anatol N. , TITLE =. J. Algebraic Combin. , FJOURNAL =. 1997 , NUMBER =. doi:10.1023/A:1008694825493 , URL =

    work page doi:10.1023/a:1008694825493 1997

  2. [2]

    arXiv e-prints , keywords =

    Skew shapes, Ehrhart positivity and beyond. arXiv e-prints , keywords =. doi:10.48550/arXiv.2503.16403 , archivePrefix =. 2503.16403 , primaryClass =

    work page doi:10.48550/arxiv.2503.16403

  3. [3]

    Liu, Fu and Tsuchiya, Akiyoshi , TITLE =. Adv. in Appl. Math. , FJOURNAL =. 2019 , PAGES =. doi:10.1016/j.aam.2019.03.004 , URL =

    work page doi:10.1016/j.aam.2019.03.004 2019

  4. [4]

    Forum Math

    Morier-Genoud, Sophie and Ovsienko, Valentin , TITLE =. Forum Math. Sigma , FJOURNAL =. 2020 , PAGES =. doi:10.1017/fms.2020.9 , URL =

    work page doi:10.1017/fms.2020.9 2020

  5. [5]

    Matroids are not. Adv. Math. , FJOURNAL =. 2022 , PAGES =. doi:10.1016/j.aim.2022.108337 , URL =

    work page doi:10.1016/j.aim.2022.108337 2022

  6. [6]

    Padrol, Arnau and Pilaud, Vincent and Ritter, Julian , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2023 , NUMBER =. doi:10.1093/imrn/rnac042 , URL =

    work page doi:10.1093/imrn/rnac042 2023

  7. [7]

    Rank polynomials of fence posets are unimodal , JOURNAL =

    Kantarc. Rank polynomials of fence posets are unimodal , JOURNAL =. 2023 , NUMBER =. doi:10.1016/j.disc.2022.113218 , URL =

    work page doi:10.1016/j.disc.2022.113218 2023

  8. [8]

    Graphs Combin

    Hibi, Takayuki and Higashitani, Akihiro and Tsuchiya, Akiyoshi and Yoshida, Koutarou , TITLE =. Graphs Combin. , FJOURNAL =. 2019 , NUMBER =. doi:10.1007/s00373-018-1990-9 , URL =

    work page doi:10.1007/s00373-018-1990-9 2019

  9. [9]

    Order Polytopes of Dimension $\leq 13$ are Ehrhart Positive

    Order Polytopes of Dimension 13 are Ehrhart Positive. arXiv e-prints , keywords =. doi:10.48550/arXiv.2412.07164 , archivePrefix =. 2412.07164 , primaryClass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2412.07164

  10. [10]

    EMS Surv

    Examples and counterexamples in Ehrhart theory. EMS Surv. Math. Sci. , FJOURNAL =. 2024 , note =

    work page 2024

  11. [11]

    Recent trends in algebraic combinatorics , SERIES =

    Liu, Fu , TITLE =. Recent trends in algebraic combinatorics , SERIES =. 2019 , MRCLASS =. doi:10.1007/978-3-030-05141-9\_6 , URL =

    work page doi:10.1007/978-3-030-05141-9 2019

  12. [12]

    , TITLE =

    McMullen, P. , TITLE =. Proc. London Math. Soc. (3) , FJOURNAL =. 1977 , NUMBER =. doi:10.1112/plms/s3-35.1.113 , URL =

    work page doi:10.1112/plms/s3-35.1.113 1977

  13. [13]

    Beck and S

    Beck, Matthias and Robins, Sinai , TITLE =. 2015 , PAGES =. doi:10.1007/978-1-4939-2969-6 , URL =

    work page doi:10.1007/978-1-4939-2969-6 2015

  14. [14]

    Sur les poly\`

    Ehrhart, Eug\`. Sur les poly\`. C. R. Acad. Sci. Paris , FJOURNAL =. 1962 , PAGES =

    work page 1962

  15. [15]

    2025 , title=

    Sam Hopkins , month =. 2025 , title=

    work page 2025

  16. [16]

    Dizier, Avery , TITLE =

    Fink, Alex and M\'esz\'aros, Karola and St. Dizier, Avery , TITLE =. Adv. Math. , FJOURNAL =. 2018 , PAGES =. doi:10.1016/j.aim.2018.05.028 , URL =

    work page doi:10.1016/j.aim.2018.05.028 2018

  17. [17]

    Fan, Neil J. Y. and Guo, Peter L. , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 2021 , PAGES =. doi:10.1016/j.jcta.2020.105311 , URL =

    work page doi:10.1016/j.jcta.2020.105311 2021

  18. [18]

    Selecta Math

    Monical, Cara and Tokcan, Neriman and Yong, Alexander , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2019 , NUMBER =. doi:10.1007/s00029-019-0513-8 , URL =

    work page doi:10.1007/s00029-019-0513-8 2019

  19. [19]

    2024 , note =

    Dania Morales , title =. 2024 , note =

    work page 2024

  20. [20]

    arXiv e-prints , keywords =

    The polytope of all matroids. arXiv e-prints , keywords =. doi:10.48550/arXiv.2502.20157 , archivePrefix =. 2502.20157 , primaryClass =

    work page doi:10.48550/arxiv.2502.20157

  21. [21]

    2024 , note=

    George Gilbert and Brian Hunt and Greta Panova , title=. 2024 , note=

    work page 2024

  22. [22]

    SIAM Journal on Discrete Mathematics , volume=

    Effective poset inequalities , author=. SIAM Journal on Discrete Mathematics , volume=. 2023 , publisher=

    work page 2023

  23. [23]

    2003 , eprint=

    A New Approach to Order Polynomials of Labeled Posets and Their Generalizations , author=. 2003 , eprint=

    work page 2003

  24. [24]

    Mathematika , volume=

    Generalized permutahedra: Minkowski linear functionals and Ehrhart positivity , author=. Mathematika , volume=. 2022 , publisher=

    work page 2022

  25. [25]

    Kahane, Yakob , title =

    work page

  26. [26]

    Kreweras, G. , year=. Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers , journal=

    work page

  27. [27]

    , TITLE =

    Speyer, David and Williams, Lauren K. , TITLE =. Trans. Amer. Math. Soc. Ser. B , FJOURNAL =. 2021 , PAGES =. doi:10.1090/btran/67 , URL =

    work page doi:10.1090/btran/67 2021

  28. [28]

    Fink, Alex and Rinc\'on, Felipe , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 2015 , PAGES =. doi:10.1016/j.jcta.2015.06.001 , URL =

    work page doi:10.1016/j.jcta.2015.06.001 2015

  29. [29]

    , TITLE =

    MacMahon, Percy A. , TITLE =. 2004 , PAGES =

    work page 2004

  30. [30]

    2023 , publisher=

    Enumerative Combinatorics , author=. 2023 , publisher=

    work page 2023

  31. [31]

    Discrete Comput

    Castillo, Federico and Liu, Fu , TITLE =. Discrete Comput. Geom. , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s00454-017-9950-3 , URL =

    work page doi:10.1007/s00454-017-9950-3 2018

  32. [32]

    Castillo, Federico and Liu, Fu , TITLE =. Algebr. Comb. , FJOURNAL =. 2021 , NUMBER =. doi:10.5802/alco.157 , URL =

    work page doi:10.5802/alco.157 2021

  33. [33]

    Mathematika , FJOURNAL =

    Jochemko, Katharina and Ravichandran, Mohan , TITLE =. Mathematika , FJOURNAL =. 2022 , NUMBER =. doi:10.1112/mtk.12122 , URL =

    work page doi:10.1112/mtk.12122 2022

  34. [34]

    Two poset polytopes

    Stanley, Richard P. , TITLE =. Discrete Comput. Geom. , FJOURNAL =. 1986 , NUMBER =. doi:10.1007/BF02187680 , URL =

    work page doi:10.1007/bf02187680 1986

  35. [35]

    Ferroni, Luis , title =. Comb. Theory , issn =. 2023 , language =. doi:10.5070/C63362796 , keywords =

    work page doi:10.5070/c63362796 2023

  36. [36]

    The unimodality of the Ehrhart $\delta$-polynomial of the chain polytope of the zig-zag poset

    The unimodality of the Ehrhart -polynomial of the chain polytope of the zig-zag poset. arXiv e-prints , keywords =. doi:10.48550/arXiv.1603.08283 , archivePrefix =. 1603.08283 , primaryClass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1603.08283

  37. [37]

    , TITLE =

    Kirillov, Anatol N. , TITLE =. Physics and combinatorics 1999 (. 2001 , ISBN =. doi:10.1142/9789812810199\_0006 , URL =

    work page doi:10.1142/9789812810199 1999

  38. [38]

    Electron

    Coons, Jane Ivy and Sullivant, Seth , TITLE =. Electron. J. Combin. , FJOURNAL =. 2023 , NUMBER =. doi:10.37236/11526 , URL =

    work page doi:10.37236/11526 2023

  39. [39]

    McGinnis, Daniel , TITLE =. J. Comb. , FJOURNAL =. 2025 , NUMBER =. doi:10.4310/joc.241216210956 , URL =

    work page doi:10.4310/joc.241216210956 2025

  40. [40]

    Bonin, Joseph and de Mier, Anna and Noy, Marc , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 2003 , NUMBER =. doi:10.1016/S0097-3165(03)00122-5 , URL =

    work page doi:10.1016/s0097-3165(03)00122-5 2003

  41. [41]

    [2023] 2023 , ISBN =

    The geometry of geometries: matroid theory, old and new , BOOKTITLE =. [2023] 2023 , ISBN =

    work page 2023

  42. [42]

    Kalai, Gil , TITLE =. I. [2023] 2023 , ISBN =. doi:10.4171/ICM2022/211 , URL =

    work page doi:10.4171/icm2022/211 2023

  43. [43]

    Eur, Christopher , TITLE =. Bull. Amer. Math. Soc. (N.S.) , FJOURNAL =. 2024 , NUMBER =. doi:10.1090/bull/1803 , URL =

    work page doi:10.1090/bull/1803 2024

  44. [44]

    Derksen, Harm and Fink, Alex , TITLE =. Adv. Math. , FJOURNAL =. 2010 , NUMBER =. doi:10.1016/j.aim.2010.04.016 , URL =

    work page doi:10.1016/j.aim.2010.04.016 2010

  45. [45]

    Valuations for matroid polytope subdivisions , JOURNAL =

    Ardila, Federico and Fink, Alex and Rinc\'. Valuations for matroid polytope subdivisions , JOURNAL =. 2010 , NUMBER =. doi:10.4153/CJM-2010-064-9 , URL =

    work page doi:10.4153/cjm-2010-064-9 2010

  46. [46]

    and de Mier, Anna , TITLE =

    Bonin, Joseph E. and de Mier, Anna , TITLE =. European J. Combin. , FJOURNAL =. 2006 , NUMBER =. doi:10.1016/j.ejc.2005.01.008 , URL =

    work page doi:10.1016/j.ejc.2005.01.008 2006

  47. [47]

    2010 , PAGES =

    Bidkhori, Hoda , TITLE =. 2010 , PAGES =

    work page 2010

  48. [48]

    Forum Math

    Lam, Thomas and Postnikov, Alexander , TITLE =. Forum Math. Sigma , FJOURNAL =. 2024 , PAGES =. doi:10.1017/fms.2024.11 , URL =

    work page doi:10.1017/fms.2024.11 2024

  49. [49]

    Chen, Yiming and Li, Yao and Yao, Ming , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2026 , NUMBER =. doi:10.1090/proc/17525 , URL =

    work page doi:10.1090/proc/17525 2026

  50. [50]

    , TITLE =

    Deligeorgaki, Danai and McGinnis, Daniel and Vindas-Mel\'endez, Andr\'es R. , TITLE =. SIAM J. Discrete Math. , FJOURNAL =. 2026 , NUMBER =. doi:10.1137/25M1738103 , URL =

    work page doi:10.1137/25m1738103 2026

  51. [51]

    Discrete Comput

    Knauer, Kolja and Mart\'inez-Sandoval, Leonardo and Ram\'irez Alfons\'in, Jorge Luis , TITLE =. Discrete Comput. Geom. , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s00454-018-9965-4 , URL =

    work page doi:10.1007/s00454-018-9965-4 2018

  52. [52]

    Kyle and Zhuang, Yan , TITLE =

    Petersen, T. Kyle and Zhuang, Yan , TITLE =. European J. Combin. , FJOURNAL =. 2025 , PAGES =. doi:10.1016/j.ejc.2024.104073 , URL =

    work page doi:10.1016/j.ejc.2024.104073 2025

  53. [53]

    Oxford Univer- sity Press (2018).https://doi.org/10.1093/oso/9780198814788.001.0001

    Oxley, James , TITLE =. 2011 , PAGES =. doi:10.1093/acprof:oso/9780198566946.001.0001 , URL =

    work page doi:10.1093/acprof:oso/9780198566946.001.0001 2011

  54. [54]

    and Smyth, Clifford , TITLE =

    McConville, Thomas and Sagan, Bruce E. and Smyth, Clifford , TITLE =. Discrete Math. , FJOURNAL =. 2021 , NUMBER =. doi:10.1016/j.disc.2021.112483 , URL =

    work page doi:10.1016/j.disc.2021.112483 2021

  55. [55]

    Ehrhart polynomials of rank two matroids , JOURNAL =

    Ferroni, Luis and Jochemko, Katharina and Schr\". Ehrhart polynomials of rank two matroids , JOURNAL =. 2022 , PAGES =. doi:10.1016/j.aam.2022.102410 , URL =

    work page doi:10.1016/j.aam.2022.102410 2022

  56. [56]

    Postnikov, Alexander , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2009 , NUMBER =. doi:10.1093/imrn/rnn153 , URL =

    work page doi:10.1093/imrn/rnn153 2009

  57. [58]

    , TITLE =

    Stanley, Richard P. , TITLE =. 2012 , PAGES =

    work page 2012

  58. [59]

    Forum Math

    Eur, Christopher and Huh, June and Larson, Matt , TITLE =. Forum Math. Pi , FJOURNAL =. 2023 , PAGES =. doi:10.1017/fmp.2023.24 , URL =

    work page doi:10.1017/fmp.2023.24 2023

  59. [60]

    Ferroni, Luis and Schr\"oter, Benjamin , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2024 , NUMBER =. doi:10.1112/jlms.12984 , URL =

    work page doi:10.1112/jlms.12984 2024

  60. [61]

    Benedetti, K

    On lattice path matroid polytopes: alcoved triangulations and snake decompositions. arXiv e-prints , keywords =. doi:10.48550/arXiv.2303.10458 , archivePrefix =. 2303.10458 , primaryClass =

    work page doi:10.48550/arxiv.2303.10458

  61. [62]

    Ehrhart polynomials of matroid polytopes and polymatroids , JOURNAL =

    De Loera, Jes\'. Ehrhart polynomials of matroid polytopes and polymatroids , JOURNAL =. 2009 , NUMBER =. doi:10.1007/s00454-008-9080-z , URL =

    work page doi:10.1007/s00454-008-9080-z 2009

  62. [63]

    Discrete Comput

    Ferroni, Luis , TITLE =. Discrete Comput. Geom. , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s00454-021-00313-4 , URL =

    work page doi:10.1007/s00454-021-00313-4 2022

  63. [64]

    Lascoux, Alain and Sch. Polyn. C. R. Acad. Sci., Paris, S. 1982 , Language =

    work page 1982

  64. [65]

    Symmetric functions,

    Manivel, Laurent , isbn =. Symmetric functions,

    work page

  65. [66]

    Macdonald, I. G. , note =

    work page

  66. [67]

    Knauer, Kolja and Mart\'inez-Sandoval, Leonardo and Ram\'irez Alfons\'in, Jorge Luis , TITLE =. Adv. in Appl. Math. , FJOURNAL =. 2018 , PAGES =. doi:10.1016/j.aam.2016.11.008 , URL =

    work page doi:10.1016/j.aam.2016.11.008 2018

  67. [68]

    Fan, Neil J. Y. and Li, Yao , TITLE =. Discrete Comput. Geom. , FJOURNAL =. 2024 , NUMBER =. doi:10.1007/s00454-023-00495-z , URL =

    work page doi:10.1007/s00454-023-00495-z 2024

  68. [69]

    and McGinnis, Daniel and Miyata, Dane and Nasr, George D

    Hanely, Derek and Martin, Jeremy L. and McGinnis, Daniel and Miyata, Dane and Nasr, George D. and Vindas-Mel\'endez, Andr\'es R. and Yin, Mei , TITLE =. Adv. Geom. , FJOURNAL =. 2023 , NUMBER =. doi:10.1515/advgeom-2023-0020 , URL =

    work page doi:10.1515/advgeom-2023-0020 2023

  69. [70]

    Macdonald, I. G. , TITLE =. Surveys in combinatorics, 1991 (. 1991 , ISBN =

    work page 1991

  70. [71]

    and Krattenthaler, C

    Gessel, Ira M. and Krattenthaler, C. , Title =. Trans. Am. Math. Soc. , ISSN =. 1997 , DOI =

    work page 1997

  71. [72]

    2024 , eprint=

    Tableau formula for vexillary double Edelman--Greene coefficients , author=. 2024 , eprint=

    work page 2024

  72. [73]

    Hopkins, Sam and Lai, Tri , Title =. J. Comb. Theory, Ser. A , ISSN =. 2021 , DOI =

    work page 2021

  73. [74]

    , Title =

    Stembridge, John R. , Title =. Adv. Math. , ISSN =. 1990 , DOI =

    work page 1990

  74. [75]

    , Title =

    Stembridge, John R. , Title =. Adv. Math. , ISSN =. 1989 , DOI =

    work page 1989

  75. [76]

    Hoffman, P. N. and Humphreys, J. F. , Title =. 1992 , Publisher =

    work page 1992

  76. [77]

    Okada, Soichi , Title =. J. Comb. Theory, Ser. A , ISSN =. 1989 , DOI =

    work page 1989

  77. [78]

    , Title =

    Stanley, Richard P. , Title =. 1986 , Language =

    work page 1986

  78. [79]

    Chan, Swee Hong and Pak, Igor and Panova, Greta , Title =. Trans. Am. Math. Soc. , ISSN =. 2022 , DOI =

    work page 2022

  79. [80]

    Linial, Nathan , Title =. SIAM J. Comput. , ISSN =. 1984 , Language =. doi:10.1137/0213049 , Keywords =

    work page doi:10.1137/0213049 1984

  80. [81]

    Kung, Joseph P. S. and Rota, Gian-Carlo and Yan, Catherine H. , title =. 2009 , publisher =

    work page 2009

Showing first 80 references.