Cycle factorizations and 1-faced graph embeddings (Q1041191)

To every factorization into transpositions of an \(n\)-cycle in the symmetric group \(S_n\), the authors assign a monomial in variables \(\omega_{ij}\) that retains the transpositions used, while forgetting their order. Summing over all possible factorizations of \(n\)-cycles, they obtain a polynomial having a closed expression and leading to a formula for the number of \(1\)-faced imbeddings of a given graph.