The Convex Hull of Parking Functions of Length $n$

Abstract

Let $\mathcal{P}_n$ be the convex hull in $\mathbb{R}^n$ of all parking functions of length $n$. Stanley found the number of vertices and the number of facets of $\mathcal{P}_n$. Building upon these results, we determine the number of faces of arbitrary dimension, the volume, and the number of integer points of $\mathcal{P}_n$.