Orientations, semiorders, arrangements, and parking functions

TL;DR

This paper generalizes the Pak-Stanley labeling to G-semiorder arrangements, establishing a bijection between regions and G-parking functions, thus extending the combinatorial understanding of hyperplane arrangements.

Contribution

It introduces G-semiorder arrangements and proves that their regions correspond to all G-parking functions, broadening the scope of hyperplane arrangement combinatorics.

Findings

Pak-Stanley labeling applies to G-semiorder arrangements

Regions of G-semiorder arrangements correspond to G-parking functions

Extends known bijection from Shi arrangements to more general arrangements

Abstract

It is known that the Pak-Stanley labeling of the Shi hyperplane arrangement provides a bijection between the regions of the arrangement and parking functions. For any graph G, we define the G-semiorder arrangement and show that the Pak-Stanley labeling of its regions produces all G-parking functions.