Generalized Parking Function Polytopes

A classical parking function of length n𝑛 is a list of positive integers (a1,a2,…,an)(𝑎1,𝑎2,…,𝑎𝑛) whose nondecreasing rearrangement b1≤b2≤⋯≤bn𝑏1≤𝑏2≤⋯≤𝑏𝑛 satisfies bi≤i𝑏𝑖≤𝑖. The convex hull of all parking functions of length n𝑛 is an n𝑛-dimensional polytope in ℝn𝑅𝑛, which we refer to as the classical parking function polytope. Its geometric properties have been explored in (Amanbayeva and Wang 2022) in response to a question posed in (Stanley 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of x𝑥-parking functions for x=(a,b,…,b)𝑥=(𝑎,𝑏,…,𝑏), which we refer to as x𝑥-parking function polytopes. We explore connections between these x𝑥-parking function polytopes, the Pitman-Stanley polytope, and the partial permutahedra of (Heuer and Striker 2022). In particular, we establish a closed-form expression for the volume of x𝑥-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. If there is time, partial progress on an extension to general x𝑥 will be presented.