Cycle factorizations and one-faced graph embeddings

Abstract

Consider factorizations into transpositions of an n-cycle in the symmetric group S_n. To every such factorization we assign a monomial in variables w_{ij} that retains the transpositions used, but forgets their order. Summing over all possible factorizations of n-cycles we obtain a polynomial that happens to admit a closed expression. From this expression we deduce a formula for the number of 1-faced embeddings of a given graph.