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.