  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.
