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| Title: | Transitive Factorizations of Permutations and Eulerian Maps in the Plane |
| Authors: | Serrano, Luis |
| Keywords: | Mathematics
combinatorics
algebra
enumeration
graph
generating function
bijection
map
ramified covers of the sphere
factorization
permutation |
| Approved Date: | 2005 |
| Date Submitted: | 2005 |
| 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. |
| Department: | Combinatorics and Optimization |
| Degree: | Master of Mathematics |
| URI: | http://hdl.handle.net/10012/1128 |
| Appears in Collections: | Electronic Theses and Dissertations (UW)
Faculty of Mathematics Theses and Dissertations
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