Transitive Factorizations of Permutations and Eulerian Maps in the Plane

dc.contributor.authorSerrano, Luisen
dc.date.accessioned2006-08-22T14:20:00Z
dc.date.available2006-08-22T14:20:00Z
dc.date.issued2005en
dc.date.submitted2005en
dc.description.abstractThe 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-M&eacute;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 <em>m</em>-Eulerian trees. In this thesis, we will give a new proof of the result by Bousquet-M&eacute;lou 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-M&eacute;lou 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.formatapplication/pdfen
dc.format.extent1153003 bytes
dc.format.mimetypeapplication/pdf
dc.identifier.urihttp://hdl.handle.net/10012/1128
dc.language.isoenen
dc.pendingfalseen
dc.publisherUniversity of Waterlooen
dc.rightsCopyright: 2005, Serrano, Luis. All rights reserved.en
dc.subjectMathematicsen
dc.subjectcombinatoricsen
dc.subjectalgebraen
dc.subjectenumerationen
dc.subjectgraphen
dc.subjectgenerating functionen
dc.subjectbijectionen
dc.subjectmapen
dc.subjectramified covers of the sphereen
dc.subjectfactorizationen
dc.subjectpermutationen
dc.titleTransitive Factorizations of Permutations and Eulerian Maps in the Planeen
dc.typeMaster Thesisen
uws-etd.degreeMaster of Mathematicsen
uws-etd.degree.departmentCombinatorics and Optimizationen
uws.peerReviewStatusUnrevieweden
uws.scholarLevelGraduateen
uws.typeOfResourceTexten

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