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The study of bipartite maps (or Grothendieck's dessins d'enfants) is closelyconnected with geometry, mathematical physics and free probability. Here westudy these objects from their permutation factorization formulation using anovel character theory approach. We first present some general symmetricfunction expressions for the number of products of two permutationsrespectively from two arbitrary, but fixed, conjugacy classes indexed by$\alpha$ and $\gamma$ which produce a permutation with $m$ cycles. Our nextobjective is to derive explicit formulas for the cases where $\alpha$corresponds to full cycles, i.e., one-face bipartite maps. We prove afar-reaching explicit formula, and show that the number for any $\gamma$ can beiteratively reduced to that of products of two full cycles, which implies anefficient dimension-reduction algorithm for building a database of all thesenumbers. Note that the number for products of two full cycles can be computedby the Zagier-Stanley formula. Also, in a unified way, we easily prove thecelebrated Harer-Zagier formula and Jackson's formula, and we obtain explicitformulas for several new families as well.

The Harer-Zagier and Jackson formulas and new results for one-face bipartite maps