Facets of the generalized Cluster complex and regions in the extended Catalan arrangement of type A

Susanna Fishel, Myrto Kallipoliti, Eleni Tzanaki
2013 unpublished
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 ∆ m (A n) and that of dominant regions in the m-Catalan arrangement Cat m (A n), where m ∈ N >0. In particular, the map which we define bijects facets containing the negative simple root −α to dominant regions having the hyperplane {v ∈ V | v, α = m} as separating wall. As a result, it restricts to a bijection between the set of facets of the positive
more » ... rt of ∆ m (A n) and the set of bounded dominant regions in 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.