The study of bipartite maps (or Grothendieck’s dessins d’enfants) is closely connected with geometry, mathematical physics and free probability. Here we study these objects from their permutation factorization formulation using a novel character theory approach. We first present some general symmetric function expressions for the number of products of two permutations respectively from two arbitrary, but fixed, conjugacy classes indexed by
α
\alpha
and
γ
\gamma
that produce a permutation with
m
m
cycles. Our next objective is to derive explicit formulas for the cases where
α
\alpha
corresponds to full cycles, i.e., one-face bipartite maps. We prove a far-reaching explicit formula, and show that the number for any
γ
\gamma
can be iteratively reduced to that of products of two full cycles, which implies an efficient dimension-reduction algorithm for building a database of all these numbers. Note that the number for products of two full cycles can be computed by the Zagier-Stanley formula. Also, in a unified way, we easily prove the celebrated Harer-Zagier formula and Jackson’s formula, and we may obtain explicit formulas for several new families as well.