Refinements of the braid arrangement and two parameter Fuss-Catalan numbers

TL;DR

This paper studies a family of hyperplane arrangements related to extended Catalan arrangements, deriving formulas for the number of regions, characteristic polynomial, and establishing bijections with decorated Dyck paths, generalizing Fuss-Catalan numbers.

Contribution

It introduces a new class of hyperplane arrangements whose regions are counted by two-parameter Fuss-Catalan numbers, and provides combinatorial and algebraic characterizations.

Findings

Number of regions given by two-parameter Fuss-Catalan numbers

Established bijection with decorated Dyck paths

Computed characteristic polynomial and interpreted coefficients

Abstract

A hyperplane arrangement in $\mathbb{R}^n$ is a finite collection of affine hyperplanes. Counting regions of hyperplane arrangements is an active research direction in enumerative combinatorics. In this paper, we consider the arrangement $\mathcal{A}_n^{(m)}$ in $\mathbb{R}^n$ given by $\{x_i=0 \mid i \in [n]\} \cup \{x_i=a^kx_j \mid k \in [-m,m], 1\leq i<j \leq n\}$ for some fixed $a>1$. It turns out that this family of arrangements is closely related to the well-studied extended Catalan arrangement of type $A$. We prove that the number of regions of $\mathcal{A}_n^{(m)}$ is a certain generalization of Catalan numbers called two parameter Fuss-Catalan numbers. We then exhibit a bijection between these regions and certain decorated Dyck paths. We also compute the characteristic polynomial and give a combinatorial interpretation for its coefficients. Most of our results also generalize to sub-arrangements of $\mathcal{A}_n^{(m)}$ by relating them to deformations of the braid arrangement.