A bijection between dominant Shi regions and core partitions

TL;DR

This paper establishes a natural bijection linking dominant regions of the m-Shi arrangement to specific core partitions, extending classical combinatorial correspondences involving Catalan numbers.

Contribution

It introduces a new bijection between m-Shi arrangement regions and (n, mn+1)-core partitions, generalizing known combinatorial relationships.

Findings

Bijection commutes with affine symmetric group actions

Extends classical Catalan number correspondences

Provides a combinatorial framework for m-Shi arrangements

Abstract

It is well-known that Catalan numbers $C_n = \frac{1}{n+1} \binom{2n}{n}$ count the number of dominant regions in the Shi arrangement of type $A$, and that they also count partitions which are both $n$-cores as well as $(n+1)$-cores. These concepts have natural extensions, which we call here the $m$-Catalan numbers and $m$-Shi arrangement. In this paper, we construct a bijection between dominant regions of the $m$-Shi arrangement and partitions which are both $n$-cores as well as $(mn+1)$-cores. The bijection is natural in the sense that it commutes with the action of the affine symmetric group.