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arXiv · 2603.24886

Bijectivity of a generalized Pak-Stanley labeling

Olivier Bernardi·Neha Goregaokar
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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.

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Olivier Bernardi, Neha Goregaokar. 2026-03-26. Bijectivity of a generalized Pak-Stanley labeling. https://arxiv.org/abs/2603.24886

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