2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/221185Consider 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.21 pages, 6 figuresCombinatoricsAlgebraic Geometrytext