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.
","abstract_html":"<p>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.</p>","abstract_has_math":false,"creators":["Daly, Daniel Alan"],"institution":null,"degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":null,"degree_department":null,"school":null,"contributors":["Petr Vojtechovsky, Ph.D.","Michael Kinyon","Richard Ball","Nicholas Galatos"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2009,"date_issued":"2009-01-01T08:00:00Z","date_published":"2009-01-01T08:00:00Z","updated_at":"2026-07-24T02:02:11Z","subjects":["Algebraic combinatorics","Coxeter groups","Permutation patterns","Algebra","Physical Sciences and Mathematics"],"languages":["en"],"rights":["Copyright is held by the author. User is responsible for all copyright compliance.
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"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Permutation Patterns, Reduced Decompositions with Few Repetitions and the Bruhat Order"]}]}],"canonical_facts":{"dc:contributor":["Petr Vojtechovsky, Ph.D.","Michael Kinyon","Richard Ball","Nicholas Galatos"],"dc:creator":["Daly, Daniel Alan"],"dc:date.available":["2001-01-01T08:00:00Z"],"dc:description.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.
"],"dc:format":["application/pdf"],"dc:identifier":["https://digitalcommons.du.edu/etd/796"],"dc:language":["en"],"dc:rights":["Copyright is held by the author. User is responsible for all copyright compliance.
"],"dc:subject":["Algebraic combinatorics","Coxeter groups","Permutation patterns","Algebra","Physical Sciences and Mathematics"],"dc:title":["Permutation Patterns, Reduced Decompositions with Few Repetitions and the Bruhat Order"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."]},"updated_at":"2026-07-24T02:02:11Z"}