Title: Generalized Parking Function Polytopes

Abstract: A classical parking function of length n is a list of positive integers (a_1, a_2, ... , a_n) whose nondecreasing rearrangement b_1 \leq b_2 \leq ... \leq b_n satisfies b_i \leq i. The convex hull of all parking functions of length n is an n-dimensional polytope in R^n, 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 x-parking functions, which we refer to as x-parking function polytopes. When x=(a,b, ... ,b), we make connections between these x-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 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. If there is time, progress on an extension to general x will be presented.