Stapledon Decompositions and Inequalities for Coefficients of Chromatic Polynomials

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.