Enumeration of planar two-face maps

Abstract

We enumerate unrooted planar maps (up to orientation-preserving homeomorphism) having two faces, according to the number of vertices and to their vertex and face degree distributions, both in the (vertex) labelled and unlabelled cases. We first consider plane maps, i.e., maps which are embedded in the plane, and then deduce the case of planar (or sphere) maps, embedded on the sphere. A crucial step is the enumeration of two-face plane maps having an antipodal symmetry and use is made of Liskovets’ method in the process. The motivation for this research comes from the topological classification of Belyi functions.