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

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 $$\{x_i=0 \mid i \in [n]\} \cup \{x_i=a^kx_j \mid k \in [-m,m], 1\le i<j \le n\}$$ 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.