# The $h^*$-polynomial of the order polytope of the zig-zag poset

## TL;DR

This paper provides a new combinatorial interpretation of the Ehrhart series coefficients for the order polytope of the zig-zag poset using shellings and the swap statistic on permutations.

## Contribution

It introduces a family of shellings for the order polytope's triangulation, linking Ehrhart series coefficients to permutation statistics.

## Key findings

- Shellings of the order polytope's triangulation are described.
- Ehrhart series coefficients are interpreted via the swap statistic.
- New combinatorial connections between polytopes and permutations are established.

## Abstract

We describe a family of shellings for the canonical triangulation of the order polytope of the zig-zag poset. This gives a new combinatorial interpretation for the coefficients in the numerator of the Ehrhart series of this order polytopein terms of the swap statistic on alternating permutations.