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

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 ${\mathrm{Cat}}^m(A_n)$, 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 ${\mathrm{Cat}}^m(A_n)$. The map ${\omega }_n$ is a composition of two bijections in which integer partitions in an $m$-staircase shape come into play.

15 pages, 6 figures

References in corpus (1)

• A uniform bijection between nonnesting and noncrossing partitions