Bijections between generalized Catalan families of types A and C

Motivated by the relation Nm(Cn)=(mn+1)Nm(An−1), holding for the m-generalized Catalan numbers of type A and C, the connection between dominant regions of the m-Shi arrangement of type An−1 and Cn 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 An−1, biject onto the set of type Cn 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×mn rectangle is provided.