  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 $\{v\in V\mid<v,\alpha >=m\}$ 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.
