Counting rooted maps on a surface

International audienceSeveral enumeration results are known about rooted maps on orientable surfaces, whereas rooted maps on non-orientable surfaces have seldom been studied. First, we unify both kind of maps, giving general functional equations for the generating series which counts rooted maps on any locally orientable surface, by number of vertices and faces. Then, we formally solve these equations, in order to establish a detailed common formula for all these generating series. All of them appear to be algebraic functions of the variables counting the number of vertices and faces. Explicit expressions and numerical tables for the series counting rooted maps on the non-orientable surfaces of genus 3 and 4 are given. (C) 2000 Elsevier Science B.V. All rights reserved