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Towards a classification of descent multiplicity-free compositions Cheek, Caleb
Abstract
This work studies combinatorics related to expansions of a quasisymmetric refinement of Schur functions into Gessel’s fundamental basis, with almost all results concerning whether an expansion is multiplicity-free. The combinatorial side of this problem concerns a certain composition poset, and whether there are two standard fillings of the same composition diagram with a given descent set. This thesis uses entirely combinatorial arguments, extending results by Bessenrodt and vanWilligenburg to work towards a classification of such descent multiplicity-free compositions. The main tools used regard the situation of appending or prepending parts to compositions. Compositions with multiplicity retain multiplicity when parts are appended or prepended, while multiplicity-free compositions stay multiplicity-free when a class of shapes called staircase-like are appended. A classification of compositions which are partitions or reverse partitions is achieved, leading up to a classification of compositions not containing a part of length one. This is used as the basis for a conjectured classification of multiplicity-free compositions without a trailing staircase. The conjecture would in turn imply a complete classification of multiplicity-free compositions.
Item Metadata
Title |
Towards a classification of descent multiplicity-free compositions
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
2012
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Description |
This work studies combinatorics related to expansions of a quasisymmetric refinement
of Schur functions into Gessel’s fundamental basis, with almost all results
concerning whether an expansion is multiplicity-free. The combinatorial side
of this problem concerns a certain composition poset, and whether there are two
standard fillings of the same composition diagram with a given descent set. This
thesis uses entirely combinatorial arguments, extending results by Bessenrodt and
vanWilligenburg to work towards a classification of such descent multiplicity-free
compositions.
The main tools used regard the situation of appending or prepending parts to
compositions. Compositions with multiplicity retain multiplicity when parts are
appended or prepended, while multiplicity-free compositions stay multiplicity-free
when a class of shapes called staircase-like are appended. A classification of compositions
which are partitions or reverse partitions is achieved, leading up to a
classification of compositions not containing a part of length one. This is used as
the basis for a conjectured classification of multiplicity-free compositions without
a trailing staircase. The conjecture would in turn imply a complete classification
of multiplicity-free compositions.
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Genre | |
Type | |
Language |
eng
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Date Available |
2012-08-30
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-ShareAlike 3.0 Unported
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DOI |
10.14288/1.0073090
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2012-11
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Campus | |
Scholarly Level |
Graduate
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-ShareAlike 3.0 Unported