A simple recurrence for covers of the sphere with branch points of arbitrary ramification

The problem of counting ramified covers of a Riemann surface up to homeomorphism was proposed by Hurwitz in the late 1800's. This problem translates combinatorially into factoring a permutation of specified cycle type, with certain conditions on the cycle types of the factors, such as minimality and transitivity. Goulden and Jackson have given a proof for the number of minimal, transitive factorizations of a permutation into transpositions. This proof involves a partial differential equation for the generating series, called the Join-Cut equation. Recently, Bousquet-M\'{e}lou and Schaeffer have found the number of minimal, transitive factorizations of a permutation into arbitrary unspecified factors. This was proved by a purely combinatorial argument, based on a direct bijection between factorizations and certain objects called $m$-Eulerian trees. In this paper, we give a simple partial differential equation for Bousquet-M\'{e}lou and Schaeffer's generating series, and for Goulden and Jackson's generating series, as well as a new proof of the result by Bousquet-M\'{e}lou and Schaeffer. We apply algebraic methods based on Lagrange's theorem, and combinatorial methods based on a new use of Bousquet-M\'{e}lou and Schaeffer's $m$-Eulerian trees.