Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS

TL;DR

This paper explores Ehrhart polynomials of order polytopes linked to finite posets, revealing combinatorial interpretations for sequences on the OEIS and connecting them to classical formulas and determinants.

Contribution

It provides new combinatorial interpretations of Ehrhart polynomials for order polytopes, linking them to OEIS sequences and classical determinant formulas.

Findings

Ehrhart polynomials relate to OEIS sequences like Catalan numbers.

Connections established between Ehrhart polynomials and classical determinant formulas.

Rediscovery of Kreweras' determinant formula via lattice path methods.

Abstract

In this paper, we provide an overview of Ehrhart polynomials associated with order polytopes of finite posets, a concept first introduced by Stanley. We focus on their combinatorial interpretations for many sequences listed on the OEIS. We begin by exploring the Ehrhart series of order polytopes resulting from various poset operations, specifically the ordinal sum and direct sum. We then concentrate on the poset $P_\lambda$ associated with the Ferrers diagram of a partition $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_t)$. When $\lambda = (k, k-1, \ldots, 1)$, the Ehrhart polynomial is a shifted Hankel determinant of the well-known Catalan numbers; when $\lambda = (k, k, \ldots, k)$, the Ehrhart polynomial is solved by Stanley's hook content formula and is used to prove conjectures for the sequence [A140934] on the OEIS. When solving these problems, we rediscover Kreweras' determinant formula for the Ehrhart polynomial $\mathrm{ehr}(\mathcal{O}(P_{\lambda}), n)$ through the application of the Lindstr\"om-Gessel-Viennot lemma on non-intersecting lattice paths.