Transitive Factorizations of Permutations and Eulerian Maps in the Plane
dc.contributor.author | Serrano, Luis | en |
dc.date.accessioned | 2006-08-22T14:20:00Z | |
dc.date.available | 2006-08-22T14:20:00Z | |
dc.date.issued | 2005 | en |
dc.date.submitted | 2005 | en |
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 <em>m</em>-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 <em>m</em>-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.identifier.uri | http://hdl.handle.net/10012/1128 | |
dc.language.iso | en | en |
dc.pending | false | 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 |
uws-etd.degree | Master of Mathematics | en |
uws-etd.degree.department | Combinatorics and Optimization | en |
uws.peerReviewStatus | Unreviewed | en |
uws.scholarLevel | Graduate | en |
uws.typeOfResource | Text | en |
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