Bijectivity of a generalized Pak-Stanley labeling

TL;DR

This paper investigates the bijectivity of a generalized Pak-Stanley labeling across various arrangements, providing conditions for when the labeling is bijective and constructing a right inverse.

Contribution

It introduces a class of arrangements where the generalized Pak-Stanley labeling is bijective and constructs a right inverse for these cases.

Findings

Identifies a class of arrangements with bijective labeling

Constructs a right inverse for the labeling in these arrangements

Shows these arrangements are the only transitive cases with bijective labeling

Abstract

The Pak-Stanley labeling is a bijection between the regions of the $m$-Shi arrangement and the $m$-parking functions. Mazin generalized this labeling to every deformation of the braid arrangement and proved that this labeling is always surjective onto a set of directed multigraph parking functions. We provide a right inverse to the generalized Pak-Stanley labeling, and identify a class $\mathcal{C}$ of arrangements for which this labeling is bijective. The class $\mathcal{C}$ includes the multi-Shi arrangements and the multi-Catalan arrangements. We also show that the arrangements in $\mathcal{C}$ are the only transitive arrangements for which the generalized Pak-Stanley labeling is bijective.