The Convex Hull of Parking Functions of Length $n$

TL;DR

This paper investigates the geometric properties of the convex hull of parking functions, providing explicit formulas for faces, volume, and lattice points, extending Stanley’s foundational results.

Contribution

It advances the understanding of the convex hull of parking functions by deriving formulas for faces, volume, and integer points, building on previous vertex and facet counts.

Findings

Derived formulas for the number of faces of all dimensions

Calculated the volume of the convex hull

Counted the integer lattice points within the convex hull

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 $\mathcal{P}_n$.