Positivity of Ehrhart polynomial coefficients

I have converted a comment (slightly modified) to the following answer as requested by Per Alexandersson. If $Ω_P(k)$ is the order polynomial of a poset $P$, then $Ω_P(k+1)$ is the Ehrhart polynomial of the order polytope $\mathcal{O}(P)$. For any n≥1, the Ehrhart polynomial of the order polytope of the poset $P_n$ with one minimal element covered by $n$ other elements is $\sum_{i=1}^{k+1}i^n$. For $n=20$ the coefficient of $k$ is $−168011/330$, so Ehrhart polynomials of 0/1 polytopes (or even order polytopes) need not have nonnegative coefficients. Incidentally, it was easy to check using Stembridge's posets package for Maple that the Ehrhart polynomial of the order polytope of any poset with at most eight elements has nonnegative coefficients.

There seems to be a lot of information (including the conjecture that this holds for the Birkhoff polytope) in Richard Stanley's slides. Of course, since Richard is a frequent participant, he can (and surely will) say more.

Fu Liu recently put on the arXiv a nice survey about this question: https://arxiv.org/abs/1711.09962v1