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PreprintCC BY 4.0arXiv

Combinatorial interpretation of the coefficients of the order polynomial of fence posets

  • Yakob Kahane
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Abstract

Given a fence poset P , we define a new statistic on permutations, denoted by blP, that provides a combinatorial interpretation of the coefficients of the order polynomial of P , answering a question of Ferroni, Morales, and Panova (2025). Using the fact that the base polytope of a lattice path matroid can be decomposed into order polytopes of fence posets, we also obtain a combinatorial interpretation of the coefficients of the Ehrhart polynomial of the base polytope of Schubert matroids, answering a question of Stanley (1999). As an application of this statistic, we establish the first nontrivial lower bound for the linear coefficient of the Ehrhart polynomial of an order polytope. Finally, we conjecture generalizations of this statistic to skew-shape posets and circular fence posets.

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Cite

BibTeXRIS
BibTeX
@misc{kahane2026combinatorial,
  title = {{Combinatorial interpretation of the coefficients of the order polynomial of fence posets}},
  author = {Yakob Kahane},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2607.11225},
  eprint = {2607.11225},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.11225},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Combinatorial interpretation of the coefficients of the order polynomial of fence posets
AU  - Yakob Kahane
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2607.11225
UR  - https://arxiv.org/abs/2607.11225
AB  - Given a fence poset P , we define a new statistic on permutations, denoted by blP, that provides a combinatorial interpretation of the coefficients of the order polynomial of P , answering a question of Ferroni, Morales, and Panova (2025). Using the fact that the base polytope of a lattice path matroid can be decomposed into order polytopes of fence posets, we also obtain a combinatorial interpretation of the coefficients of the Ehrhart polynomial of the base polytope of Schubert matroids, answering a question of Stanley (1999). As an application of this statistic, we establish the first nontrivial lower bound for the linear coefficient of the Ehrhart polynomial of an order polytope. Finally, we conjecture generalizations of this statistic to skew-shape posets and circular fence posets.
ER  -