Are there any results stating that a given family of convex polytopes have Ehrhart polynomials with non-negative coefficients?
What methods are available for proving such a property for some family of polytopes?
Remember, the Ehrhart polynomial $p(k)$ for a convex polytope $P$ with integer vertices is given by the property that $p(k)$ counts the number of integer lattice points in the $k$-dilation of $P$, where $k$ is a positive integer.
Best Answer
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.