Order Polytopes of Dimension $\leq 13$ are Ehrhart Positive

TL;DR

This paper proves that all order polytopes of dimension 12 or 13 are Ehrhart positive, resolving an open problem and extending known results up to dimension 11.

Contribution

It confirms Ehrhart positivity for order polytopes of dimension 12 and 13, filling the gap between previous results and counterexamples for higher dimensions.

Findings

Order polytopes of dimension 12 and 13 are Ehrhart positive.

All h*-polynomials of order polytopes in these dimensions are real-rooted.

This resolves an open problem on Ehrhart positivity in these dimensions.

Abstract

The order polytopes arising from the finite poset were first introduced and studied by Stanley. For any positive integer $d\geq 14$, Liu and Tsuchiya proved that there exists a non-Ehrhart positive order polytope of dimension $d$. They also proved that any order polytope of dimension $d\leq 11$ is Ehrhart positive. We confirm that any order polytope of dimension $12$ or $13$ is Ehrhart positive. This solves an open problem proposed by Liu and Tsuchiya. Besides, we also verify that any $h^{*}$-polynomial of order polytope of dimension $d\leq 13$ is real-rooted.