Ehrhart polynomials of partial permutohedra
For positive integers $m$ and $n$, the partial permutohedron $\mathcal{P}(m,n)$ is a certain integral polytope in $\mathbb{R}^m$, which can be defined as the convex hull of the vectors from $\{0,1,\ldots,n\}^m$ whose nonzero entries are distinct. For $n=m-1$, $\mathcal{P}(m,m-1)$ is (after translation by $(1,\ldots,1)$) the polytope $P_m$ of parking functions of length $m$, and for $n\ge m$, $\mathcal{P}(m,n)$ is combinatorially equivalent to an $m$-stellohedron. The main result of this paper is an explicit expression for the Ehrhart polynomial of $\mathcal{P}(m,n)$ for any $m$ and $n$ with $n\ge m-1$. The result confirms the validity of a conjecture for this Ehrhart polynomial in arXiv:2207.14253, and the $n=m-1$ case also answers a question of Stanley regarding the number of integer points in $P_m$. The proof of the result involves transforming $\mathcal{P}(m,n)$ to a unimodularly equivalent polytope in $\mathbb{R}^{m+1}$, obtaining a decomposition of this lifted version of $\mathcal{P}(m,n)$ with $n\ge m-1$ as a Minkowski sum of dilated coordinate simplices, applying a result of Postnikov for the number of integer points in generalized permutohedra of this form, observing that this gives an expression for the Ehrhart polynomial of $\mathcal{P}(m,n)$ with $n\ge m-1$ as an edge-weighted sum over graphs (with loops and multiple edges permitted) on $m$ labelled vertices in which each connected component contains at most one cycle, and then applying standard techniques for the enumeration of such graphs.