A note on irreducible maps with several boundaries

TL;DR

This paper derives a formula for counting d-irreducible bipartite planar maps with multiple boundaries, extending previous formulas and providing explicit counts for special cases using a tree-based interpretation.

Contribution

It extends Collet and Fusy's formula to include multiple boundaries and irreducibility constraints, offering explicit enumeration formulas for various classes of bipartite planar maps.

Findings

Derived a general generating function formula for d-irreducible bipartite planar maps with boundaries.

Provided explicit enumeration formulas for maps without multiple edges, 4-irreducible maps, and girth at least 6.

Extended previous work to cover maps with multiple marked faces and irreducibility conditions.

Abstract

We derive a formula for the generating function of d-irreducible bipartite planar maps with several boundaries, i.e. having several marked faces of controlled degrees. It extends a formula due to Collet and Fusy for the case of arbitrary (non necessarily irreducible) bipartite planar maps, which we recover by taking d=0. As an application, we obtain an expression for the number of d-irreducible bipartite planar maps with a prescribed number of faces of each allowed degree. Very explicit expressions are given in the case of maps without multiple edges (d=2), 4-irreducible maps and maps of girth at least 6 (d=4). Our derivation is based on a tree interpretation of the various encountered generating functions.