A simple recurrence for covers of the sphere with branch points of arbitrary ramification
| dc.creator | Goulden, I. P. | |
| dc.creator | Serrano, Luis G. | |
| dc.date | 2005-08-12 | |
| dc.date.accessioned | 2026-07-07T05:22:20Z | |
| dc.date.available | 2026-07-07T05:22:20Z | |
| dc.description | 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-Mélou 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-Mélou and Schaeffer's generating series, and for Goulden and Jackson's generating series, as well as a new proof of the result by Bousquet-Mélou and Schaeffer. We apply algebraic methods based on Lagrange's theorem, and combinatorial methods based on a new use of Bousquet-Mélou and Schaeffer's $m$-Eulerian trees. | |
| dc.identifier | https://arxiv.org/abs/math/0508226 | |
| dc.identifier | http://arxiv.org/abs/math/0508226 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/76022 | |
| dc.subject | Combinatorics | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 05A15 (Primary) 14N10 (Secondary) | |
| dc.title | A simple recurrence for covers of the sphere with branch points of arbitrary ramification | |
| dc.type | text |