A Simple Formula for the Series of Constellations and Quasi-constellations with Boundaries

We obtain a very simple formula for the generating function of bipartite (resp. quasi-bipartite) planar maps with boundaries (holes) of prescribed lengths, which generalizes certain expressions obtained by Eynard in a book to appear. The formula is derived from a bijection due to Bouttier, Di Francesco and Guitter combined with a process (reminiscent of a construction of Pitman) of aggregating connected components of a forest into a single tree. The formula naturally extends to p-constellations and quasi-p-constellations with boundaries (the case p = 2 corresponding to bipartite maps).