Efficient enumeration of rooted maps of a given orientable genus by number of faces and vertices

We simplify the recurrence satisfied by the polynomial part of the generating function that counts rooted maps of positive orientable genus g by number of vertices and faces. We have written an optimized program in C++ for computing this generating function and constructing tables of numbers of rooted maps, and we describe some of these optimizations here. Using this program we extended the enumeration of rooted maps of orientable genus g by number of vertices and faces to g = 4, 5 and 6 and by number of edges to g = 5 and 6 and conjectured a further simplification of the generating function that counts rooted maps by number of edges. Our program is documented and available on request, allowing anyone with a sufficiently powerful computer to carry the calculations even further.