Factoring N-Cycles and Counting Maps of Given Genus

Abstract

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 ℓ(λ) + ℓ(μ) = n + 1 - 2g where ℓ(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.