Facets of the r-stable n,k-hypersimplex

TL;DR

This paper characterizes the facets of r-stable n,k-hypersimplices, determines their count, and explores conditions under which they are Gorenstein, revealing new classes with unimodal Ehrhart δ-vectors.

Contribution

It explicitly determines the facets of r-stable hypersimplices and identifies infinite Gorenstein families, expanding understanding of their geometric and combinatorial properties.

Findings

r-stable n,k-hypersimplices have 2n facets for r<⌊n/k⌋

Identifies infinite Gorenstein r-stable hypersimplices

Expands known classes with unimodal Ehrhart δ-vectors

Abstract

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.