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

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.