Skew shapes, Ehrhart positivity, and beyond

Abstract

A classical result by Kreweras (1965) allows one to compute the number of plane partitions of a given skew shape and bounded parts as certain determinants. We prove that these determinants expand as polynomials with nonnegative coefficients. This result can be reformulated in terms of order polynomials of cell posets of skew shapes and explains important positivity phenomena about the Ehrhart polynomials of shard polytopes, matroids, and order polytopes. Among other applications, we generalize a positivity statement from Schubert calculus by Fomin and Kirillov (1997) from straight shapes to skew shapes. We show that all shard polytopes are Ehrhart positive and, stronger, that all fence posets, including the zig‐zag poset, and all circular fence posets have order polynomials with nonnegative coefficients. We discuss a general method for proving positivity that reduces to showing positivity of the linear terms of the order polynomials. We propose positivity conjectures on other relevant classes of posets.

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Cite

BibTeX RIS

BibTeX

@article{ferroni2026skew,
  title = {{Skew shapes, Ehrhart positivity, and beyond}},
  author = {Luis Ferroni and Alejandro H. Morales and Greta Panova},
  journal = {Proceedings of the London Mathematical Society},
  year = {2026},
  month = aug,
  doi = {10.1112/plms.70204},
  url = {https://doi.org/10.1112/plms.70204},
  note = {CC BY},
}

RIS

TY  - JOUR
TI  - Skew shapes, Ehrhart positivity, and beyond
AU  - Luis Ferroni
AU  - Alejandro H. Morales
AU  - Greta Panova
PY  - 2026
DA  - 2026/08/01
JO  - Proceedings of the London Mathematical Society
DO  - 10.1112/plms.70204
UR  - https://doi.org/10.1112/plms.70204
AB  - A classical result by Kreweras (1965) allows one to compute the number of plane partitions of a given skew shape and bounded parts as certain determinants. We prove that these determinants expand as polynomials with nonnegative coefficients. This result can be reformulated in terms of order polynomials of cell posets of skew shapes and explains important positivity phenomena about the Ehrhart polynomials of shard polytopes, matroids, and order polytopes. Among other applications, we generalize a positivity statement from Schubert calculus by Fomin and Kirillov (1997) from straight shapes to skew shapes. We show that all shard polytopes are Ehrhart positive and, stronger, that all fence posets, including the zig‐zag poset, and all circular fence posets have order polynomials with nonnegative coefficients. We discuss a general method for proving positivity that reduces to showing positivity of the linear terms of the order polynomials. We propose positivity conjectures on other relevant classes of posets.
SN  - 0024-6115
SN  - 1460-244X
ER  -

Abstract

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