Bijections for faces of the Shi and Catalan arrangements

TL;DR

This paper provides a bijective proof linking faces of Shi and Catalan arrangements to decorated binary trees and functions, extending previous formulas and offering new combinatorial insights.

Contribution

It introduces a novel bijection for faces of the Shi arrangement for any dimension, connecting them to decorated binary trees and functions, and extends these results to Catalan arrangements.

Findings

First bijective proof of Athanasiadis's face formula

Bijection between faces and decorated binary trees

Counting formulas for faces of Catalan arrangements

Abstract

In 1986, Shi derived the famous formula $(n+1)^{n-1}$ for the number of regions of the Shi arrangement, a hyperplane arrangement in $\mathbb{R}^n$. There are at least two different bijective explanations of this formula, one by Pak and Stanley, another by Athanasiadis and Linusson. In 1996, Athanasiadis used the finite field method to derive a formula for the number of $k$-dimensional faces of the Shi arrangement for any $k$. Until now, the formula of Athanasiadis did not have a bijective explanation. In this paper, we extend a bijection for regions defined by Bernardi to obtain a bijection between the $k$-dimensional faces of the Shi arrangement for any $k$ and a set of decorated binary trees. Furthermore, we show how these trees can be converted to a simple set of functions of the form $f: [n-1] \to [n+1]$ together with a marked subset of $\text{Im}(f)$. This correspondence gives the first bijective proof of the formula of Athanasiadis. In the process, we also obtain a bijection and counting formula for the faces of the Catalan arrangement. All of our results generalize to both extended arrangements.