Séminaire Lotharingien de Combinatoire, 95B.93 (2026), 12 pp.
Olivier Bernardi and Neha Goregaokar
Bijectivity of a Generalized Pak-Stanley 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 C of arrangements for which this labeling is
bijective. The class C includes the multi-Shi arrangements and
the multi-Catalan arrangements. We also show that the arrangements
in C are the only transitive arrangements for which the
generalized Pak-Stanley labeling is bijective.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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