Factoring N-Cycles and Counting Maps of Given Genus

We present an explicit expression for the number of decompositions of an n−cycle as a product of any two permutations of cycle types given by partitions λ and μ. The same expression is also counting the number of unicellular rooted bicolored maps on an orientable surface of genus g with vertex degree distribution given by λ and μ. The relation between the genus and the partitions λ and μ is given by `(λ) + `(μ) = n + 1 − 2g where `(λ) is the number of parts of λ. We use character theory and the group algebra of the symmetric group to develop our expression. The key argument is the construction of a bijection involving the character formula at one end and our final expression at the other end.