Generalized Parking Function Polytopes

A classical parking function of length n is a list of positive integers \((a_1, a_2, \ldots , a_n)\) whose nondecreasing rearrangement \(b_1 \le b_2 \le \cdots \le b_n\) satisfies \(b_i \le i\). The convex hull of all parking functions of length n is an n-dimensional polytope in \({\mathbb {R}}^n\), which we refer to as the classical parking function polytope. Its geometric properties have been explored in Amanbayeva and Wang (Enumer Combin Appl 2(2):Paper No. S2R10, 10, 2022) in response to a question posed by Stanley (Amer Math Mon 127(6):563–571, 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of \({\textbf{x}}\)-parking functions for \({\textbf{x}}=(a,b,\dots ,b)\), which we refer to as \({\textbf{x}}\)-parking function polytopes. We explore connections between these \({\textbf{x}}\)-parking function polytopes, the Pitman–Stanley polytope, and the partial permutahedra of Heuer and Striker (SIAM J Discrete Math 36(4):2863–2888, 2022). In particular, we establish a closed-form expression for the volume of \({\textbf{x}}\)-parking function polytopes. This allows us to answer a conjecture of Behrend et al. (2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary.