Posets, parking functions and the regions of the Shi arrangement revisited

TL;DR

This paper revisits the enumeration of regions in Shi arrangements of types A and C, providing explicit bijections with sequences and posets, extending previous combinatorial proofs and descriptions.

Contribution

It describes the Athanasiadis-Linusson bijection and generalizes it to type C arrangements, linking regions to sequences and posets with new combinatorial insights.

Findings

Bijection between type A_{n-1} Shi arrangement regions and sequences of length n.

Extension of bijections to type C arrangements with sequences over a symmetric integer set.

Posets encoding regions and their antichains provide a combinatorial framework.

Abstract

The number of regions of the type A_{n-1} Shi arrangement in R^n is counted by the intrinsically beautiful formula (n+1)^{n-1}. First proved by Shi, this result motivated Pak and Stanley as well as Athanasiadis and Linusson to provide bijective proofs. We give a description of the Athanasiadis-Linusson bijection and generalize it to a bijection between the regions of the type C_n Shi arrangement in R^n and sequences a_1a_2...a_n, where a_i \in \{-n, -n+1,..., -1, 0, 1,..., n-1, n\}, i \in [n]. Our bijections naturally restrict to bijections between regions of the arrangements with a certain number of ceilings (or floors) and sequences with a given number of distinct elements. A special family of posets, whose antichains encode the regions of the arrangements, play a central role in our approach.