A simple formula for the series of constellations and quasi-constellations with boundaries

TL;DR

This paper derives a simple generating function formula for bipartite and quasi-bipartite planar maps with boundaries, extending previous results and utilizing bijections and aggregation processes.

Contribution

It introduces a new, simplified formula for generating functions of constellations with boundaries, generalizing earlier expressions and extending to p-constellations.

Findings

Derived a simple formula for bipartite planar maps with boundaries

Extended the formula to p-constellations and quasi-p-constellations

Connected the formula to bijections and forest aggregation processes

Abstract

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).