On decompositions of a cyclic permutation into a product of a given number of permutations

The investigation of decompositions of a permutation into a product of permutations satisfying certain conditions plays a key role in the study of meromorphic functions or, equivalently, branched coverings of the 2-sphere; it goes back to A. Hurwitz' work in the late nineteenth century. In 2000 M. Bousquet-Melou and G. Schaeffer obtained an elegant formula for the number of decompositions of a permutation into a product of a given number of permutations corresponding to coverings of genus 0. Their formula has not been generalized to coverings of the sphere by surfaces of higher genera so far. This paper contains a new proof of the Bousquet-Melou–Schaeffer formula for the case of decompositions of a cyclic permutation, which, hopefully, can be generalized to positive genera.