Date of Award

1-1-2009

Document Type

Dissertation

Degree Name

Ph.D.

Organizational Unit

College of Natual Science and Mathematics

First Advisor

Petr Vojtechovsky, Ph.D.

Second Advisor

Michael Kinyon

Third Advisor

Richard Ball

Fourth Advisor

Nicholas Galatos

Keywords

Algebraic combinatorics, Coxeter groups, Permutation patterns

Abstract

This thesis is concerned with problems involving permutations. The main focus is on connections between permutation patterns and reduced decompositions with few repetitions. Connections between permutation patterns and reduced decompositions were first studied various mathematicians including Stanley, Billey and Tenner. In particular, they studied pattern avoidance conditions on reduced decompositions with no repeated elements. This thesis classifies the pattern avoidance and containment conditions on reduced decompositions with one and two elements repeated. This classification is then used to obtain new enumeration results for pattern classes related to the reduced decompositions and introduces the technique of counting pattern classes via reduced decompositions. In particular, counts on pattern classes involving 1 or 2 copies of the patterns 321 and 3412 are obtained. Pattern conditions are then used to classify and enumerate downsets in the Bruhat order for the symmetric group and the rook monoid which is a generalization of the symmetric group. Finally, motivated by coding theory, the concepts of displacement, additive stretch and multiplicative stretch of permutations are introduced. These concepts are then analyzed with respect to maximality and distribution as a new prospect for improving interleaver design.

Publication Statement

Copyright is held by the author. User is responsible for all copyright compliance.

Rights Holder

Daniel Alan Daly

Provenance

Received from ProQuest

File Format

application/pdf

Language

en

File Size

162 p.

Discipline

Mathematics

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