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.