Ehrhart positivity for marked order polytopes

TL;DR

This paper establishes criteria for Ehrhart positivity of marked order polytopes, extending previous results and confirming conjectures on positivity for various classes of polytopes.

Contribution

It provides a new criterion for Ehrhart coefficient nonnegativity and proves Ehrhart positivity for marked order polytopes of skew shapes, confirming several conjectures.

Findings

Marked order polytopes of skew shapes are Ehrhart positive.

Extended Ehrhart positivity results to skew Gelfand-Tsetlin and m-generalized Pitman-Stanley polytopes.

Provided a criterion for nonnegativity of polynomial coefficients in this context.

Abstract

Given a pair of finite posets $A \subseteq P$, the function counting integer-valued order preserving extensions of an order preserving map $\lambda : A\rightarrow \mathbb{Z}$ from $A$ to $P$ is given by a piecewise polynomial in $\lambda$. We provide a criterion for the nonnegativity of the coefficients of these multivariate polynomials and apply it to show that marked order polytopes of skew shapes are Ehrhart positive in a multivariate sense. This extends recent results of Ferroni-Morales-Panova on order polytopes of skew shapes and proves conjectures on the Ehrhart positivity of skew Gelfand-Tsetlin polytopes and $m$-generalized Pitman-Stanley polytopes due to Alexanderson-Alhajjar and Dugan-Hegarty-Morales-Raymond, respectively.