Transitive Factorizations of Permutations and Eulerian Maps in the Plane
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 min...
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ndltd-LACETR-oai-collectionscanada.gc.ca-OWTU.10012-11282013-10-04T04:07:18ZSerrano, Luis2006-08-22T14:20:00Z2006-08-22T14:20:00Z20052005http://hdl.handle.net/10012/1128The 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.application/pdf1153003 bytesapplication/pdfenUniversity of WaterlooCopyright: 2005, Serrano, Luis. All rights reserved.Mathematicscombinatoricsalgebraenumerationgraphgenerating functionbijectionmapramified covers of the spherefactorizationpermutationTransitive Factorizations of Permutations and Eulerian Maps in the PlaneThesis or DissertationCombinatorics and OptimizationMaster of Mathematics |
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Mathematics combinatorics algebra enumeration graph generating function bijection map ramified covers of the sphere factorization permutation |
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Mathematics combinatorics algebra enumeration graph generating function bijection map ramified covers of the sphere factorization permutation Serrano, Luis Transitive Factorizations of Permutations and Eulerian Maps in the Plane |
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 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. |
author |
Serrano, Luis |
author_facet |
Serrano, Luis |
author_sort |
Serrano, Luis |
title |
Transitive Factorizations of Permutations and Eulerian Maps in the Plane |
title_short |
Transitive Factorizations of Permutations and Eulerian Maps in the Plane |
title_full |
Transitive Factorizations of Permutations and Eulerian Maps in the Plane |
title_fullStr |
Transitive Factorizations of Permutations and Eulerian Maps in the Plane |
title_full_unstemmed |
Transitive Factorizations of Permutations and Eulerian Maps in the Plane |
title_sort |
transitive factorizations of permutations and eulerian maps in the plane |
publisher |
University of Waterloo |
publishDate |
2006 |
url |
http://hdl.handle.net/10012/1128 |
work_keys_str_mv |
AT serranoluis transitivefactorizationsofpermutationsandeulerianmapsintheplane |
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1716599582279860224 |