Stanley's non-Ehrhart-positive order polytopes

TL;DR

This paper investigates the Ehrhart positivity of a specific family of order polytopes, providing explicit formulas for their Ehrhart coefficients and demonstrating the existence of non-Ehrhart-positive polytopes in all dimensions above 20.

Contribution

It offers explicit formulas for Ehrhart coefficients of a family of order polytopes and answers open questions about the existence of non-Ehrhart-positive polytopes in various dimensions.

Findings

Existence of non-Ehrhart-positive order polytopes in all dimensions ≥21.

Construction of order polytopes with any number of negative Ehrhart coefficients.

Explicit formulas for Ehrhart coefficients in terms of Bernoulli numbers.

Abstract

We say a polytope is Ehrhart positive if all the coefficients in its Ehrhart polynomial are positive. Answering an Ehrhart positivity question posed on Mathoverflow, Stanley provided an example of a non-Ehrhart-positive order polytope of dimension $21$. Stanley's example comes from a certain family of order polytopes. In this paper, we study the Ehrhart positivity question on this family of polytopes. By giving explicit formulas for the coefficients of the Ehrhart polynomials of these polytopes in terms of Bernolli numbers, we determine the sign of each Ehrhart coefficient of each polytope in the family. As a consequence of our result, we conclude that for any positive integer $d \ge 21,$ there exists an order polytope of dimension $d$ that is not Ehrhart positive, and for any positive integer $\ell$, there exists an order polytope whose Ehrhart polynomial has precisely $\ell$ negative coefficients, which answers a question posed by Hibi. We finish this article by discussing the existence of lower-dimensional order polytopes whose Ehrhart polynomials have a negative coefficient.