Parking Function Polytopes

TL;DR

This paper explores the geometric and combinatorial properties of parking function polytopes, introducing new combinatorial structures and deriving formulas for their $h$-polynomials, volumes, and Ehrhart polynomials, while connecting them to other polytopes.

Contribution

It introduces binary and skewed binary partitions to analyze parking function polytopes and provides explicit formulas for their $h$-polynomials and volume calculations.

Findings

Explicit formula for $h$-polynomials in terms of generalized Eulerian polynomials.

Characterization of normal fans via skewed binary partitions.

Formulas for volumes and Ehrhart polynomials of parking function polytopes.

Abstract

We extend the notion of parking function polytopes and study their geometric and combinatorial structure, including normal fans, face posets, and $h$-polynomials, as well as their connections to other classes of polytopes. To capture their combinatorial features, we introduce generalizations of ordered set partitions, called binary partitions and skewed binary partitions. Using properties of preorder cones, we characterize the skewed binary partitions that are in bijection with the cones of the normal fan of a parking function polytope. This description of the normal fan yields an explicit formula for the $h$-polynomials of simple parking function polytopes in terms of generalized Eulerian polynomials. Finally, we relate parking function polytopes to several well-known polytopes, leading to additional results, including formulas for their volumes and Ehrhart polynomials.