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arXiv · 1302.2441

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

Susanna Fishel·Myrto Kallipoliti·Eleni Tzanaki
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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 $\{v\in V\mid =m\}$ 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.

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Susanna Fishel, Myrto Kallipoliti, Eleni Tzanaki. 2013-02-11. Facets of the $m$-generalized cluster complex and regions in the $m$-extended Catalan arrangement of type $A_n$. https://arxiv.org/abs/1302.2441

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