Bijections between generalized Catalan families of types A and C

Motivated by the relation ${\mathsf{N}}^m\left(C_n\right)=(mn+1){\mathsf{N}}^m\left(A_{n?1}\right)$, holding for the $m$-generalized Catalan numbers of type $A$ and $C$, the connection between dominant regions of the $m$-Shi arrangement of type $A_{n?1}$ and $C_n$ is investigated. More precisely, it is explicitly shown how $mn+1$ copies of the set of dominant regions of the $m$-Shi arrangement of type $A_{n?1}$, biject onto the set of type $C_n$ such regions. This is achieved by exploiting two different viewpoints of the representative alcove of each region: the Shi tableau and the abacus diagram. In the same line of thought, a bijection between $mn+1$ copies of the set of $m$-Dyck paths of height $n$ and the set of $N?E$ lattice paths inside an $n\times mn$ rectangle is provided.