Cycle factorizations and 1-faced graph embeddings

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 i j 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.