Bijections on $r$-Shi and $r$-Catalan Arrangements

TL;DR

This paper introduces a cubic matrix-based framework to establish bijections between regions of $r$-Shi and $r$-Catalan arrangements and combinatorial objects like $r$-trees, permutation-Dyck path pairs, and parking functions.

Contribution

It presents a unified method using cubic matrices to derive multiple bijections connecting geometric arrangements with combinatorial structures.

Findings

Bijection from $r$-Shi regions to $O$-rooted labeled $r$-trees.

Bijection from $r$-Catalan regions to permutation and $r$-Dyck path pairs.

Recovery of Pak-Stanley labeling via positive entries in row slices.

Abstract

Associated with the $r$-Shi arrangement and $r$-Catalan arrangement in $\Bbb{R}^n$, we introduce a cubic matrix for each region to establish two bijections in a uniform way. Firstly, the positions of minimal positive entries in column slices of the cubic matrix will give a bijection from regions of the $r$-Shi arrangement to $O$-rooted labeled $r$-trees. Secondly, the numbers of positive entries in column slices of the cubic matrix will give a bijection from regions of the $r$-Catalan arrangement to pairings of permutation and $r$-Dyck path. Moreover, the numbers of positive entries in row slices of the cubic matrix will recover the Pak-Stanley labeling, a celebrated bijection from regions of the $r$-Shi arrangement to $r$-parking functions.