A Note on Irreducible Maps with Several Boundaries

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 is recovered 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.