The genus series for maps Original Research Article

Two-cell embeddings of graphs in orientable surfaces have been studied extensively by combinatorialists, prompted partly by the four-colour problem. The genus series remains difficult to determine except in a number of special cases. An important integral representation has been obtained by Bessis et al. (1980) and ʹt Hooft (1974). We show that this representation follows, surprisingly, by a finitary, and therefore combinatorial, argument from a characterisation of embeddings as permutations and from classical properties of symmetric functions. In principle, this argument makes these techniques accessible to other combinatorial constructions.