A famous and wide-open problem, going back to at least the early 1970's, concerns the classification of chromatic polynomials of graphs. Toward this classification problem, one may ask for necessary inequalities among the coefficients of a chromatic polynomial, and we contribute one such set of inequalities when a chromatic polynomial $\chi_G(n) = \chi^*_0 \binom {n+d} d + \chi^*_1 \binom {n+d-1} d + \dots + \chi^*_d \binom n d$ is written in terms of a binomial-coefficient basis. More precisely, we prove that $\chi^*_{d-2}+\chi^*_{d-3}+\dots+\chi^*_{d-j-1} \ \ge \ \chi^*_1+\chi^*_2+\dots+\chi^*_j $, for $1 \le j \le \lfloor \frac{ d }{ 2 } \rfloor - 1$. A similar result holds for flow polynomials enumerating either modular or integral nowhere-zero flows of a graph. Our theorems follow from connections among chromatic, flow, order, and Ehrhart polynomials, and the fact that the latter satisfy a decomposition formula into symmetric polynomials due to Stapledon.

(This is joint work with Emerson Le\'on.)