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

Abstract

The order polytopes arising from the finite poset were first introduced and studied by Stanley. For any positive integer $d\ge 14$, 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 $d\le 11$ 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 $h^{\ast }$-polynomial of order polytope of dimension $d\le 13$ is real-rooted.

7 pages, Comments are welcome

References in corpus (2)

• Stanley's non-Ehrhart-positive order polytopes
• Polytopes and large counterexamples