Enumeration of simple bipartite maps on the sphere and the projective plane (Q5957755)

The maps considered in this paper are simple (without loops or parallel edges), bipartite and rooted (with a distinguished oriented edge, the root). An explicit formula is given for the number of such maps with \(m\) edges on the plane, and functional equations are given for the generating functions that count these maps, both on the plane and the projective plane, by the number of edges, the valency of the root-face and the number of non-root faces. As corollaries, the enumerating functions are given for those maps that have one or two faces, on both the plane and the projective plane.