Facets of the Generalized Cluster Complex and Regions in the Extended Catalan Arrangement of Type A.

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 Cat(m)(A(n)), where m is an element of N->0. In particular, the map which we define bijects facets containing the negative simple root -alpha to dominant regions having the hyperplane {v is an element of V vertical bar < v, alpha > = 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 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.