Set-Valued Young Tableaux and Product-Coproduct Prographs
Standard set-valued Young tableaux are a generalization of standard Young tableaux where cells can contain unordered sets of integers, with the added condition that every integer at position $(i,j)$ must be smaller that every integer at both $(i+1,j)$ and $(i,j+1)$. In this paper, we explore propert...
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creator | Drube, Paul Krueger, Maxwell Skalsky, Ashley Wren, Meghan |
description | Standard set-valued Young tableaux are a generalization of standard Young
tableaux where cells can contain unordered sets of integers, with the added
condition that every integer at position $(i,j)$ must be smaller that every
integer at both $(i+1,j)$ and $(i,j+1)$. In this paper, we explore properties
of standard set-valued Young tableaux with three rows and a fixed number of
integers in every cell of each row (referred to as set-valued tableaux with
row-constant density). Our primary focus is on standard set-valued Young
tableaux with $1$ integer in each first-row cell, $k-1$ integers in each
second-row cell, and $1$ integer in each third-row cell. For rectangular shapes
$\lambda=n^3$, such tableaux are placed in bijection with closed $k$-ary
product-coproduct prographs: directed plane graphs that correspond to finite
compositions involving a $k$-ary product operator and a $k$-ary coproduct
operator. That bijection is extended to three-row set-valued Young tableaux of
non-rectangular and skew shape, and it is shown that a set-valued analogue of
the Sch\"utzenberger involution on tableaux corresponds to $180$-degree
rotation of the associated prographs. As a set-valued analogue of the
hook-length formula is currently lacking, we also present direct enumerations
of three-row standard set-valued Young tableaux for a variety of row-constant
densities and a small number of columns. We then argue why the numbers of
tableaux with the row-constant density $(1,k-1,1)$ should be interpreted as a
one-parameter generalization of the three-dimensional Catalan numbers that
mirrors the generalization of the (two-dimensional) Catalan numbers provided by
the $k$-Catalan numbers. |
doi_str_mv | 10.48550/arxiv.1710.02709 |
format | Article |
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tableaux where cells can contain unordered sets of integers, with the added
condition that every integer at position $(i,j)$ must be smaller that every
integer at both $(i+1,j)$ and $(i,j+1)$. In this paper, we explore properties
of standard set-valued Young tableaux with three rows and a fixed number of
integers in every cell of each row (referred to as set-valued tableaux with
row-constant density). Our primary focus is on standard set-valued Young
tableaux with $1$ integer in each first-row cell, $k-1$ integers in each
second-row cell, and $1$ integer in each third-row cell. For rectangular shapes
$\lambda=n^3$, such tableaux are placed in bijection with closed $k$-ary
product-coproduct prographs: directed plane graphs that correspond to finite
compositions involving a $k$-ary product operator and a $k$-ary coproduct
operator. That bijection is extended to three-row set-valued Young tableaux of
non-rectangular and skew shape, and it is shown that a set-valued analogue of
the Sch\"utzenberger involution on tableaux corresponds to $180$-degree
rotation of the associated prographs. As a set-valued analogue of the
hook-length formula is currently lacking, we also present direct enumerations
of three-row standard set-valued Young tableaux for a variety of row-constant
densities and a small number of columns. We then argue why the numbers of
tableaux with the row-constant density $(1,k-1,1)$ should be interpreted as a
one-parameter generalization of the three-dimensional Catalan numbers that
mirrors the generalization of the (two-dimensional) Catalan numbers provided by
the $k$-Catalan numbers.</description><identifier>DOI: 10.48550/arxiv.1710.02709</identifier><language>eng</language><subject>Mathematics - Combinatorics</subject><creationdate>2017-10</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,778,883</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/1710.02709$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.1710.02709$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Drube, Paul</creatorcontrib><creatorcontrib>Krueger, Maxwell</creatorcontrib><creatorcontrib>Skalsky, Ashley</creatorcontrib><creatorcontrib>Wren, Meghan</creatorcontrib><title>Set-Valued Young Tableaux and Product-Coproduct Prographs</title><description>Standard set-valued Young tableaux are a generalization of standard Young
tableaux where cells can contain unordered sets of integers, with the added
condition that every integer at position $(i,j)$ must be smaller that every
integer at both $(i+1,j)$ and $(i,j+1)$. In this paper, we explore properties
of standard set-valued Young tableaux with three rows and a fixed number of
integers in every cell of each row (referred to as set-valued tableaux with
row-constant density). Our primary focus is on standard set-valued Young
tableaux with $1$ integer in each first-row cell, $k-1$ integers in each
second-row cell, and $1$ integer in each third-row cell. For rectangular shapes
$\lambda=n^3$, such tableaux are placed in bijection with closed $k$-ary
product-coproduct prographs: directed plane graphs that correspond to finite
compositions involving a $k$-ary product operator and a $k$-ary coproduct
operator. That bijection is extended to three-row set-valued Young tableaux of
non-rectangular and skew shape, and it is shown that a set-valued analogue of
the Sch\"utzenberger involution on tableaux corresponds to $180$-degree
rotation of the associated prographs. As a set-valued analogue of the
hook-length formula is currently lacking, we also present direct enumerations
of three-row standard set-valued Young tableaux for a variety of row-constant
densities and a small number of columns. We then argue why the numbers of
tableaux with the row-constant density $(1,k-1,1)$ should be interpreted as a
one-parameter generalization of the three-dimensional Catalan numbers that
mirrors the generalization of the (two-dimensional) Catalan numbers provided by
the $k$-Catalan numbers.</description><subject>Mathematics - Combinatorics</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotj81qwkAUhWfjomgfoKvmBcbecTJzM8sS-gdCBYPQVbh3fqyQmjCaYt--Vbs6h29xOJ8QdwrmZWUMPFA-7b7nCv8ALBDcjXDreJQb6sYYio9-3G-LhriLNJ4K2odilfsw-qOs--HazmSbafg8zMQkUXeIt_85Fc3zU1O_yuX7y1v9uJRk0UkmTkpTaSEyozYQSKHXCh2jNZU3ZL3ToeLko1r4UgOXYBhTCMoiJD0V99fZy_d2yLsvyj_t2aG9OOhf6qZBhA</recordid><startdate>20171007</startdate><enddate>20171007</enddate><creator>Drube, Paul</creator><creator>Krueger, Maxwell</creator><creator>Skalsky, Ashley</creator><creator>Wren, Meghan</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20171007</creationdate><title>Set-Valued Young Tableaux and Product-Coproduct Prographs</title><author>Drube, Paul ; Krueger, Maxwell ; Skalsky, Ashley ; Wren, Meghan</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a679-babf13a460ebb7350da17c3179b7658c5a6c93d8bfce12c430b405b7fdd1670f3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Mathematics - Combinatorics</topic><toplevel>online_resources</toplevel><creatorcontrib>Drube, Paul</creatorcontrib><creatorcontrib>Krueger, Maxwell</creatorcontrib><creatorcontrib>Skalsky, Ashley</creatorcontrib><creatorcontrib>Wren, Meghan</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Drube, Paul</au><au>Krueger, Maxwell</au><au>Skalsky, Ashley</au><au>Wren, Meghan</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Set-Valued Young Tableaux and Product-Coproduct Prographs</atitle><date>2017-10-07</date><risdate>2017</risdate><abstract>Standard set-valued Young tableaux are a generalization of standard Young
tableaux where cells can contain unordered sets of integers, with the added
condition that every integer at position $(i,j)$ must be smaller that every
integer at both $(i+1,j)$ and $(i,j+1)$. In this paper, we explore properties
of standard set-valued Young tableaux with three rows and a fixed number of
integers in every cell of each row (referred to as set-valued tableaux with
row-constant density). Our primary focus is on standard set-valued Young
tableaux with $1$ integer in each first-row cell, $k-1$ integers in each
second-row cell, and $1$ integer in each third-row cell. For rectangular shapes
$\lambda=n^3$, such tableaux are placed in bijection with closed $k$-ary
product-coproduct prographs: directed plane graphs that correspond to finite
compositions involving a $k$-ary product operator and a $k$-ary coproduct
operator. That bijection is extended to three-row set-valued Young tableaux of
non-rectangular and skew shape, and it is shown that a set-valued analogue of
the Sch\"utzenberger involution on tableaux corresponds to $180$-degree
rotation of the associated prographs. As a set-valued analogue of the
hook-length formula is currently lacking, we also present direct enumerations
of three-row standard set-valued Young tableaux for a variety of row-constant
densities and a small number of columns. We then argue why the numbers of
tableaux with the row-constant density $(1,k-1,1)$ should be interpreted as a
one-parameter generalization of the three-dimensional Catalan numbers that
mirrors the generalization of the (two-dimensional) Catalan numbers provided by
the $k$-Catalan numbers.</abstract><doi>10.48550/arxiv.1710.02709</doi><oa>free_for_read</oa></addata></record> |
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subjects | Mathematics - Combinatorics |
title | Set-Valued Young Tableaux and Product-Coproduct Prographs |
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