Facets of the r-stable n,k-hypersimplex

Takayuki Hibi, Liam Solus

Abstract: Let k,n and r be positive integers with k<n and r≤⌊nk⌋. 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<⌊nk⌋. 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 δ-vectors.