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 α and γ γ that produce a permutation with m m cycles. Our next objective is to derive explicit formulas for the cases where α α 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 γ 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.
No takes yet. Share an insight, caveat, or question.
R.-J. Chen (2024) studied this question.