Stapledon Decompositions and Inequalities for Coefficients of Chromatic Polynomials

TL;DR

This paper applies Stapledon’s polynomial decomposition to Ehrhart and chromatic polynomials, revealing nonnegative coefficient decompositions that lead to new inequalities for chromatic polynomial coefficients.

Contribution

It introduces a novel decomposition approach for Ehrhart and chromatic polynomials, establishing inequalities for chromatic polynomial coefficients in simple graphs.

Findings

Decomposition of Ehrhart series numerator polynomial into symmetric polynomials

Decomposition for chromatic polynomial numerator polynomial

Derivation of inequalities for chromatic polynomial coefficients

Abstract

We use a polynomial decomposition result by Stapledon to show that the numerator polynomial of the Ehrhart series of an open polytope is the difference of two symmetric polynomials with nonnegative integer coefficients. We obtain a related decomposition for order polytopes and for the numerator polynomial of the corresponding series for chromatic polynomials. The nonnegativity of the coefficients in such decompositions provide inequalities satisfied by the coefficients of chromatic polynomials for any simple graph.