Facets of the Generalized Cluster Complex and Regions in the Extended Catalan Arrangement of Type $A$
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 ${\rm Cat}^m(A_n)$, where $m\in\mathbb{N}_{>0}$. In particular, the map which we define bijects facets containing the negative …