Geodesic
Facets of the generalized cluster complex and regions in the extended Catalan arrangement of type \(A\)
Susanna Fishel1 ; Myrto Kallipoliti2 ; Eleni Tzanaki3
1Arizona State University
2University of Vienna
3University of Crete
The electronic journal of combinatorics, Tome 20 (2013) no. 4
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In this paper we present a bijection between two well known families of Catalan objects: the set of facets of the $m$-generalized cluster complex $\Delta^m(A_n)$ and that of dominant regions in the $m$-Catalan arrangement ${\rm Cat}^m(A_n)$, where $m\in\mathbb{N}_{>0}$. In particular, the map which we define bijects facets containing the negative simple root $-\alpha$ to dominant regions having the hyperplane $\{v\in V\mid\left\langle v,\alpha \right\rangle=m\}$ as separating wall. As a result, it 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 ${\rm Cat}^m(A_n)$. Our map is a composition of two bijections in which integer partitions in an $m$-dilated $n$-staircase shape come into play.
DOI : 10.37236/3169
Classification : 05A17, 05E45, 11P81, 52C35
Mots-clés : cluster complex, Catalan arrangements, integer partitions

Susanna Fishel  1   ; Myrto Kallipoliti  2   ; Eleni Tzanaki  3

1 Arizona State University
2 University of Vienna
3 University of Crete 
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     title = {Facets of the generalized cluster complex and regions in the extended {Catalan} arrangement of type {\(A\)}},
     journal = {The electronic journal of combinatorics},
     year = {2013},
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Susanna Fishel; Myrto Kallipoliti; Eleni Tzanaki. Facets of the generalized cluster complex and regions in the extended Catalan arrangement of type \(A\). The electronic journal of combinatorics, Tome 20 (2013) no. 4. doi: 10.37236/3169

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