The unimodality of the Ehrhart $\delta$-polynomial of the chain polytope of the zig-zag poset

TL;DR

This paper proves the unimodality of the Ehrhart δ-polynomial for the chain polytope of the zig-zag poset, confirming a conjecture by Kirillov through connections to W-polynomials and known unimodality results.

Contribution

It establishes the unimodality of the Ehrhart δ-polynomial for a specific class of polytopes, linking it to W-polynomials and leveraging existing unimodality theorems.

Findings

Ehrhart δ-polynomial coincides with the W-polynomial for the zig-zag poset.

Unimodality of the δ-polynomial follows from known properties of W-polynomials.

Confirms Kirillov's conjecture on the unimodality of this polynomial.

Abstract

We prove the unimodality of the Ehrhart $\delta$-polynomial of the chain polytope of the zig-zag poset, which was conjectured by Kirillov. First, based on a result due to Stanley, we show that this polynomial coincides with the $W$-polynomial for the zig-zag poset with some natural labeling. Then, its unimodality immediately follows from a result of Gasharov, which states that the $W$-polynomials of naturally labeled graded posets of rank $1$ or $2$ are unimodal.