dc.contributor.author Serrano, Luis en dc.date.accessioned 2006-08-22 14:20:00 (GMT) dc.date.available 2006-08-22 14:20:00 (GMT) dc.date.issued 2005 en dc.date.submitted 2005 en dc.identifier.uri http://hdl.handle.net/10012/1128 dc.description.abstract 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 with a 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. Furthermore, this argument is generalized to surfaces of higher genus. 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 thesis, we will give a new proof of the result by Bousquet-Mélou and Schaeffer, introducing a simple partial differential equation. 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. Some partial results are also given for a refinement of this problem, in which the number of cycles in each factor is specified. This involves Lagrange's theorem in many variables. en dc.format application/pdf en dc.format.extent 1153003 bytes dc.format.mimetype application/pdf dc.language.iso en en dc.publisher University of Waterloo en dc.rights Copyright: 2005, Serrano, Luis. All rights reserved. en dc.subject Mathematics en dc.subject combinatorics en dc.subject algebra en dc.subject enumeration en dc.subject graph en dc.subject generating function en dc.subject bijection en dc.subject map en dc.subject ramified covers of the sphere en dc.subject factorization en dc.subject permutation en dc.title Transitive Factorizations of Permutations and Eulerian Maps in the Plane en dc.type Master Thesis en dc.pending false en uws-etd.degree.department Combinatorics and Optimization en uws-etd.degree Master of Mathematics en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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