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Experimental methods in permutation patterns and bijective proof

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TitleInfo
Title
Experimental methods in permutation patterns and bijective proof
Name (type = personal)
NamePart (type = family)
Shar
NamePart (type = given)
Nathaniel
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Nathaniel Shar
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author
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Zeilberger
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Doron
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Doron Zeilberger
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Advisory Committee
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chair
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Sloane
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Neil
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Neil Sloane
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Advisory Committee
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internal member
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Retakh
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Vladimir
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Vladimir Retakh
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Advisory Committee
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internal member
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Pudwell
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Lara
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Lara Pudwell
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Advisory Committee
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Rutgers University
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degree grantor
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Graduate School - New Brunswick
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Text
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theses
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DateCreated (qualifier = exact)
2016
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2016-05
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2016
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xx
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eng
Subject (authority = RUETD)
Topic
Mathematics
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Title
Rutgers University Electronic Theses and Dissertations
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ETD_7165
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electronic resource
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Note
Supplementary File: Words123
Extent
1 online resource (vi, 102 p. : ill.)
Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references
Subject (authority = ETD-LCSH)
Topic
Combinatorial enumeration problems
Abstract (type = abstract)
Experimental mathematics is the technique of developing conjectures and proving theorems through the use of experimentation; that is, exploring finitely many cases and detecting patterns that can then be rigorously proved. This thesis applies the techniques of experimental mathematics to several problems.
First, we generalize the translation method of Wood and Zeilberger [49] to algebraic proofs, and as an example, produce (by computer) the first bijective proof of Franel’s recurrence for a<sub>n</sub><sup>(3)</sup>=&Sigma;<sup>n</sup><sub>k=0</sub>(<sup>n</sup><sub>k</sub>)<sup>3</sup>.
Next, we apply the method of enumeration schemes to several problems in the fieldof patterns on permutations and words. Given a word w on the alphabet [n] and σ ∈ S<sub>k</sub>, we say that w contains the pattern σ if some subsequence of the letters of w is orderisomorphic to σ. First, we find an enumeration scheme that allows us to count the words containing r copies of each letter that avoid the pattern 123. Then we look at the case where w is in fact a permutation in S<sub>n</sub>. A repeating permutation is one that is the direct sum of several copies of a smaller permutation. We produce an enumeration scheme to count permutations avoiding repeating patterns of low codimension, and show that for each repeating pattern, the problem belongs to the eventually polynomial ansatz.
Note (type = statement of responsibility)
by Nathaniel Shar
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Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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NjNbRU
Identifier (type = doi)
doi:10.7282/T3BK1FKF
Genre (authority = ExL-Esploro)
ETD doctoral
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The author owns the copyright to this work.
RightsHolder (type = personal)
Name
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Shar
GivenName
Nathaniel
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Type
Permission or license
DateTime (encoding = w3cdtf); (qualifier = exact); (point = start)
2016-04-11 15:52:48
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Nathaniel Shar
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Copyright holder
Affiliation
Rutgers University. Graduate School - New Brunswick
AssociatedObject
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Author Agreement License
Detail
I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
Copyright
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Copyright protected
Availability
Status
Open
Reason
Permission or license
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