Ehrhart h ∗-Vectors of Hypersimplices

We consider the Ehrhart h[superscript ∗]-vector for the hypersimplex. It is well-known that the sum of the h[superscript ∗][subscript i] is the normalized volume which equals the Eulerian numbers. The main result is a proof of a conjecture by R. Stanley which gives an interpretation of the h[superscript ∗][subscript i] coefficients in terms of descents and exceedances. Our proof is geometric using a careful book-keeping of a shelling of a unimodular triangulation. We generalize this result to other closely related polytopes.National Science Foundation (U.S.) (Grant DMS-0604423

Facets of the r-stable n,k-hypersimplex

Let $k, n$ and $r$ be positive integers with $k < n$ and $r\leq\lfloor\frac{n}{k}\rfloor$. We determine the facets of the $r$-stable $n,k$-hypersimplex. As a result, it turns out that the $r$-stable $n,k$-hypersimplex has exactly $2n$ facets for every $r<\lfloor\frac{n}{k}\rfloor$. We then utilize the equations of the facets to study when the $r$-stable hypersimplex is Gorenstein. For every $k>0$ we identify an infinite collection of Gorenstein $r$-stable hypersimplices, consequently expanding the collection of $r$-stable hypersimplices known to have unimodal Ehrhart $\delta$-vectors.Comment: 12 pages, 2 figure

Unimodality Problems in Ehrhart Theory

Ehrhart theory is the study of sequences recording the number of integer points in non-negative integral dilates of rational polytopes. For a given lattice polytope, this sequence is encoded in a finite vector called the Ehrhart $h^*$-vector. Ehrhart $h^*$-vectors have connections to many areas of mathematics, including commutative algebra and enumerative combinatorics. In this survey we discuss what is known about unimodality for Ehrhart $h^*$-vectors and highlight open questions and problems.Comment: Published in Recent Trends in Combinatorics, Beveridge, A., et al. (eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27. This version updated October 2017 to correct an error in the original versio