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Minimal Transitive Factorizations of Permutations into Cycles

Published online by Cambridge University Press:  20 November 2018

John Irving
John Irving*
Affiliation:
Department of Mathematics and Computing Science, Saint Mary’s University, Halifax, NS, B3H 3C3, e-mail: john.irving@smu.ca
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Abstract

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We introduce a new approach to an enumerative problem closely linked with the geometry of branched coverings, that is, we study the number ${{H}_{\alpha }}({{i}_{2}},{{i}_{3}},...)$ of ways a given permutation (with cycles described by the partition $\alpha $) can be decomposed into a product of exactly ${{i}_{2}}$ 2-cycles, ${{i}_{3}}$ 3-cycles, etc., with certain minimality and transitivity conditions imposed on the factors. The method is to encode such factorizations as planar maps with certain descent structure and apply a new combinatorial decomposition to make their enumeration more manageable. We apply our technique to determine ${{H}_{\alpha }}({{i}_{2}},{{i}_{3}},...)$ when $\alpha $ has one or two parts, extending earlier work of Goulden and Jackson. We also show how these methods are readily modified to count inequivalent factorizations, where equivalence is defined by permitting commutations of adjacent disjoint factors. Our technique permits us to generalize recent work of Goulden, Jackson, and Latour, while allowing for a considerable simplification of their analysis.

Keywords

05A1505E10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 5 , 01 October 2009 , pp. 1092 - 1117
Copyright
Copyright © Canadian Mathematical Society 2009

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