Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, Baxter, and related monoids
Autor(a) principal: | |
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Data de Publicação: | 2019 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | https://doi.org/10.1016/j.jalgebra.2019.06.025 |
Resumo: | The cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift. This paper examines the cyclic shift graphs of ‘plactic-like’ monoids, whose elements can be viewed as combinatorial objects of some type: aside from the plactic monoid itself (the monoid of Young tableaux), examples include the hypoplactic monoid (quasi-ribbon tableaux), the sylvester monoid (binary search trees), the stalactic monoid (stalactic tableaux), the taiga monoid (binary search trees with multiplicities), and the Baxter monoid (pairs of twin binary search trees). It was already known that for many of these monoids, connected components of the cyclic shift graph consist of elements that have the same evaluation (that is, contain the same number of each generating symbol). This paper focuses on the maximum diameter of a connected component of the cyclic shift graph of these monoids in the rank-n case. For the hypoplactic monoid, this is n−1; for the sylvester and taiga monoids, at least n−1 and at most n; for the stalactic monoid, 3 (except for ranks 1 and 2, when it is respectively 0 and 1); for the plactic monoid, at least n−1 and at most 2n−3. The current state of knowledge, including new and previously-known results, is summarized in a table. |
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Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, Baxter, and related monoidsBinary search treeConjugacyCyclageCyclic shiftHypoplactic monoidPlactic monoidQuasi-ribbon tableauSylvester monoidAlgebra and Number TheoryThe cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift. This paper examines the cyclic shift graphs of ‘plactic-like’ monoids, whose elements can be viewed as combinatorial objects of some type: aside from the plactic monoid itself (the monoid of Young tableaux), examples include the hypoplactic monoid (quasi-ribbon tableaux), the sylvester monoid (binary search trees), the stalactic monoid (stalactic tableaux), the taiga monoid (binary search trees with multiplicities), and the Baxter monoid (pairs of twin binary search trees). It was already known that for many of these monoids, connected components of the cyclic shift graph consist of elements that have the same evaluation (that is, contain the same number of each generating symbol). This paper focuses on the maximum diameter of a connected component of the cyclic shift graph of these monoids in the rank-n case. For the hypoplactic monoid, this is n−1; for the sylvester and taiga monoids, at least n−1 and at most n; for the stalactic monoid, 3 (except for ranks 1 and 2, when it is respectively 0 and 1); for the plactic monoid, at least n−1 and at most 2n−3. The current state of knowledge, including new and previously-known results, is summarized in a table.CMA - Centro de Matemática e AplicaçõesDM - Departamento de MatemáticaRUNCain, Alan J.Malheiro, António2019-07-18T22:51:42Z2019-10-012019-10-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article66application/pdfhttps://doi.org/10.1016/j.jalgebra.2019.06.025eng0021-8693PURE: 14143367http://www.scopus.com/inward/record.url?scp=85068467706&partnerID=8YFLogxKhttps://doi.org/10.1016/j.jalgebra.2019.06.025info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2024-03-11T04:34:41Zoai:run.unl.pt:10362/75892Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T03:35:35.288412Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, Baxter, and related monoids |
title |
Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, Baxter, and related monoids |
spellingShingle |
Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, Baxter, and related monoids Cain, Alan J. Binary search tree Conjugacy Cyclage Cyclic shift Hypoplactic monoid Plactic monoid Quasi-ribbon tableau Sylvester monoid Algebra and Number Theory |
title_short |
Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, Baxter, and related monoids |
title_full |
Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, Baxter, and related monoids |
title_fullStr |
Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, Baxter, and related monoids |
title_full_unstemmed |
Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, Baxter, and related monoids |
title_sort |
Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, Baxter, and related monoids |
author |
Cain, Alan J. |
author_facet |
Cain, Alan J. Malheiro, António |
author_role |
author |
author2 |
Malheiro, António |
author2_role |
author |
dc.contributor.none.fl_str_mv |
CMA - Centro de Matemática e Aplicações DM - Departamento de Matemática RUN |
dc.contributor.author.fl_str_mv |
Cain, Alan J. Malheiro, António |
dc.subject.por.fl_str_mv |
Binary search tree Conjugacy Cyclage Cyclic shift Hypoplactic monoid Plactic monoid Quasi-ribbon tableau Sylvester monoid Algebra and Number Theory |
topic |
Binary search tree Conjugacy Cyclage Cyclic shift Hypoplactic monoid Plactic monoid Quasi-ribbon tableau Sylvester monoid Algebra and Number Theory |
description |
The cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift. This paper examines the cyclic shift graphs of ‘plactic-like’ monoids, whose elements can be viewed as combinatorial objects of some type: aside from the plactic monoid itself (the monoid of Young tableaux), examples include the hypoplactic monoid (quasi-ribbon tableaux), the sylvester monoid (binary search trees), the stalactic monoid (stalactic tableaux), the taiga monoid (binary search trees with multiplicities), and the Baxter monoid (pairs of twin binary search trees). It was already known that for many of these monoids, connected components of the cyclic shift graph consist of elements that have the same evaluation (that is, contain the same number of each generating symbol). This paper focuses on the maximum diameter of a connected component of the cyclic shift graph of these monoids in the rank-n case. For the hypoplactic monoid, this is n−1; for the sylvester and taiga monoids, at least n−1 and at most n; for the stalactic monoid, 3 (except for ranks 1 and 2, when it is respectively 0 and 1); for the plactic monoid, at least n−1 and at most 2n−3. The current state of knowledge, including new and previously-known results, is summarized in a table. |
publishDate |
2019 |
dc.date.none.fl_str_mv |
2019-07-18T22:51:42Z 2019-10-01 2019-10-01T00:00:00Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://doi.org/10.1016/j.jalgebra.2019.06.025 |
url |
https://doi.org/10.1016/j.jalgebra.2019.06.025 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
0021-8693 PURE: 14143367 http://www.scopus.com/inward/record.url?scp=85068467706&partnerID=8YFLogxK https://doi.org/10.1016/j.jalgebra.2019.06.025 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
66 application/pdf |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
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Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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