Degrees of interior polynomials and parking function enumerators

TL;DR

This paper explores the degrees of interior polynomials and parking function enumerators in directed graphs and oriented matroids, providing formulas and generalizations that connect combinatorial properties with polytope geometry.

Contribution

It introduces formulas for the degree of interior polynomials and parking function enumerators, extending results to oriented regular matroids and arbitrary rooted directed graphs.

Findings

Degree of interior polynomial linked to minimum directed join size

Formula for leading coefficient of the interior polynomial

Degree of parking function enumerator related to feedback arc sets

Abstract

The interior polynomial of a directed graph is defined as the $h^*$-polynomial of the graph's (extended) root polytope, and it displays several attractive properties. Here we express its degree in terms of the minimum cardinality of a directed join, and give a formula for the leading coefficient. We present natural generalizations of these results to oriented regular matroids; in the process we also give a facet description for the extended root polytope of an oriented regular matroid. By duality, our expression for the degree of the interior polynomial implies a formula for the degree of the parking function enumerator of an Eulerian directed graph (which is equivalent to the greedoid polynomial of the corresponding branching greedoid). We extend that result to obtain the degree of the parking function enumerator of an arbitrary rooted directed graph in terms of the minimum cardinality of a certain type of feedback arc set.