Bijectivity of a generalized Pak-Stanley labeling
arXiv:2603.24886
Abstract
The Pak-Stanley labeling is a bijection between the regions of the -Shi arrangement and the -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 of arrangements for which this labeling is bijective. The class includes the multi-Shi arrangements and the multi-Catalan arrangements. We also show that the arrangements in are the only transitive arrangements for which the generalized Pak-Stanley labeling is bijective.
14 pages, 6 figures. Extended abstract accepted to FPSAC 2026