  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 such 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. For example, we show that $\chi^*_{ j } \le
\chi^*_{ d-j }$, for $0 \le j \le \frac{ d }{ 2 }$. Similar results hold for
flow and tension polynomials enumerating either modular or integral
nowhere-zero flows/tensions of a graph. Our theorems follow from connections
among chromatic, flow, tension, and order polynomials, as well as Ehrhart
polynomials of lattice polytopes that admit unimodular triangulations. Our
results use Ehrhart inequalities due to Athanasiadis and Stapledon and are
related to recent work by Hersh--Swartz and Breuer--Dall, where inequalities
similar to some of ours were derived using algebraic-combinatorial methods.
