Facets of the $m$-generalized cluster complex and regions in the $m$-extended Catalan arrangement of type $A_n$

TL;DR

This paper establishes a bijection between facets of the $m$-generalized cluster complex and regions in the $m$-extended Catalan arrangement of type $A_n$, linking combinatorial and geometric structures.

Contribution

It introduces a new bijection connecting facets of the $m$-generalized cluster complex with dominant regions in the $m$-Catalan arrangement, involving integer partitions.

Findings

Bijection between facets and regions established

Facets containing a negative simple root correspond to regions with specific hyperplanes

Bijection involves $m$-staircase shape partitions

Abstract

In this paper we present a bijection $\omega_n$ between two well known families of Catalan objects: the set of facets of the $m$-generalized cluster complex $\Delta^m(A_n)$ and the set of dominant regions in the $m$-Catalan arrangement ${\rm Cat}^m(A_n)$, where $m\in\mathbb{N}_{>0}$. In particular, $\omega_n$ bijects the facets containing the negative simple root $-\alpha$ to dominant regions having the hyperplane $\{v\in V\mid<v,\alpha >=m\}$ as separating wall. As a result, $\omega_n$ restricts to a bijection between the set of facets of the positive part of $\Delta^m(A_n)$ and the set of bounded dominant regions in ${\rm Cat}^m(A_n)$. The map $\omega_n$ is a composition of two bijections in which integer partitions in an $m$-staircase shape come into play.