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

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 $\alpha$ and $\gamma$ which produce a permutation with $m$ cycles. Our next objective is to derive explicit formulas for the cases where $\alpha$ corresponds to full cycles, i.e., one-face bipartite maps. We prove a far-reaching explicit formula, and show that the number for any $\gamma$ 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 obtain explicit formulas for several new families as well.