{"id":"https://openalex.org/W3025444033","doi":"https://doi.org/10.37236/9481","title":"Commutation Classes of the Reduced Words for the Longest Element of ${\\mathfrak S}_n$","display_name":"Commutation Classes of the Reduced Words for the Longest Element of ${\\mathfrak S}_n$","publication_year":2020,"publication_date":"2020-05-15","ids":{"openalex":"https://openalex.org/W3025444033","doi":"https://doi.org/10.37236/9481","mag":"3025444033"},"language":"en","primary_location":{"id":"doi:10.37236/9481","is_oa":true,"landing_page_url":"https://doi.org/10.37236/9481","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i2p21/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i2p21/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5017189560","display_name":"Gon\u00e7alo Gutierres","orcid":"https://orcid.org/0000-0001-9480-498X"},"institutions":[{"id":"https://openalex.org/I76903346","display_name":"University of Coimbra","ror":"https://ror.org/04z8k9a98","country_code":"PT","type":"education","lineage":["https://openalex.org/I76903346"]}],"countries":["PT"],"is_corresponding":true,"raw_author_name":"Gon\u00e7alo Gutierres","raw_affiliation_strings":["University of Coimbra"],"affiliations":[{"raw_affiliation_string":"University of Coimbra","institution_ids":["https://openalex.org/I76903346"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5073618674","display_name":"Ricardo Mamede","orcid":"https://orcid.org/0000-0002-9264-6604"},"institutions":[{"id":"https://openalex.org/I76903346","display_name":"University of Coimbra","ror":"https://ror.org/04z8k9a98","country_code":"PT","type":"education","lineage":["https://openalex.org/I76903346"]}],"countries":["PT"],"is_corresponding":false,"raw_author_name":"Ricardo Mamede","raw_affiliation_strings":["University of Coimbra"],"affiliations":[{"raw_affiliation_string":"University of Coimbra","institution_ids":["https://openalex.org/I76903346"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100608390","display_name":"Jos\u00e9 Lu\u00eds Santos","orcid":"https://orcid.org/0000-0002-2727-6774"},"institutions":[{"id":"https://openalex.org/I76903346","display_name":"University of Coimbra","ror":"https://ror.org/04z8k9a98","country_code":"PT","type":"education","lineage":["https://openalex.org/I76903346"]}],"countries":["PT"],"is_corresponding":false,"raw_author_name":"Jos\u00e9 Luis Santos","raw_affiliation_strings":["University of Coimbra, CMUC, Department of Mathematics Portugal"],"affiliations":[{"raw_affiliation_string":"University of Coimbra, CMUC, Department of Mathematics Portugal","institution_ids":["https://openalex.org/I76903346"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5017189560"],"corresponding_institution_ids":["https://openalex.org/I76903346"],"apc_list":null,"apc_paid":null,"fwci":1.0998,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.72715934,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"27","issue":"2","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9894000291824341,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9894000291824341,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11419","display_name":"Supramolecular Self-Assembly in Materials","score":0.9521999955177307,"subfield":{"id":"https://openalex.org/subfields/2502","display_name":"Biomaterials"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10207","display_name":"Advanced biosensing and bioanalysis techniques","score":0.9366000294685364,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8527556657791138},{"id":"https://openalex.org/keywords/bijection","display_name":"Bijection","score":0.818588137626648},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8173251152038574},{"id":"https://openalex.org/keywords/element","display_name":"Element (criminal law)","score":0.5496297478675842},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.5104295611381531},{"id":"https://openalex.org/keywords/symmetric-group","display_name":"Symmetric group","score":0.43986907601356506},{"id":"https://openalex.org/keywords/coxeter-group","display_name":"Coxeter group","score":0.4378608167171478},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3506048619747162}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8527556657791138},{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.818588137626648},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8173251152038574},{"id":"https://openalex.org/C200288055","wikidata":"https://www.wikidata.org/wiki/Q2621792","display_name":"Element (criminal law)","level":2,"score":0.5496297478675842},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5104295611381531},{"id":"https://openalex.org/C128622974","wikidata":"https://www.wikidata.org/wiki/Q849512","display_name":"Symmetric group","level":2,"score":0.43986907601356506},{"id":"https://openalex.org/C143669375","wikidata":"https://www.wikidata.org/wiki/Q7874246","display_name":"Coxeter group","level":2,"score":0.4378608167171478},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3506048619747162},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.37236/9481","is_oa":true,"landing_page_url":"https://doi.org/10.37236/9481","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i2p21/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:estudogeral.sib.uc.pt:10316/101354","is_oa":true,"landing_page_url":"http://hdl.handle.net/10316/101354","pdf_url":null,"source":{"id":"https://openalex.org/S4306401208","display_name":"Estudo Geral (Universidade de Coimbra)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2802025818","host_organization_name":"Hospitais da Universidade de Coimbra","host_organization_lineage":["https://openalex.org/I2802025818"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nd","license_id":"https://openalex.org/licenses/cc-by-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:estudogeral.uc.pt:10316/101354","is_oa":true,"landing_page_url":"https://hdl.handle.net/10316/101354","pdf_url":null,"source":{"id":"https://openalex.org/S7407055320","display_name":"Estudo Geral","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.37236/9481","is_oa":true,"landing_page_url":"https://doi.org/10.37236/9481","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i2p21/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1499778340","display_name":null,"funder_award_id":"UIDB/00324/2020","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G213682523","display_name":null,"funder_award_id":"FCT/MCTES","funder_id":"https://openalex.org/F4320323642","funder_display_name":"Minist\u00e9rio da Ci\u00eancia, Tecnologia e Ensino Superior"},{"id":"https://openalex.org/G3272457656","display_name":null,"funder_award_id":"UIDB/","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G4941397794","display_name":null,"funder_award_id":"UIDB/00324/2020","funder_id":"https://openalex.org/F4320323392","funder_display_name":"Universidade de Coimbra"},{"id":"https://openalex.org/G5226653915","display_name":null,"funder_award_id":"FCT/MCT","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G738030663","display_name":null,"funder_award_id":"FCT/MCTES","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"}],"funders":[{"id":"https://openalex.org/F4320323392","display_name":"Universidade de Coimbra","ror":"https://ror.org/04z8k9a98"},{"id":"https://openalex.org/F4320323642","display_name":"Minist\u00e9rio da Ci\u00eancia, Tecnologia e Ensino Superior","ror":"https://ror.org/045b9pr88"},{"id":"https://openalex.org/F4320334779","display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","ror":"https://ror.org/00snfqn58"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3025444033.pdf","grobid_xml":"https://content.openalex.org/works/W3025444033.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W112671161","https://openalex.org/W753498374","https://openalex.org/W1569820777","https://openalex.org/W1965541390","https://openalex.org/W1985268811","https://openalex.org/W1986929557","https://openalex.org/W1995154714","https://openalex.org/W2029428693","https://openalex.org/W2053480094","https://openalex.org/W2229099293","https://openalex.org/W2281505443","https://openalex.org/W2755434175","https://openalex.org/W2963900142","https://openalex.org/W3102410742","https://openalex.org/W4246894044"],"related_works":["https://openalex.org/W1517690850","https://openalex.org/W2810190866","https://openalex.org/W3041986789","https://openalex.org/W4297094178","https://openalex.org/W2086531994","https://openalex.org/W3206431649","https://openalex.org/W1974527570","https://openalex.org/W2952682049","https://openalex.org/W4300771154","https://openalex.org/W3025444033"],"abstract_inverted_index":{"Using":[0],"the":[1,6,14,32,49,76,81,86,92],"standard":[2,139],"Coxeter":[3],"presentation":[4],"for":[5,13,85,105],"symmetric":[7,93],"group":[8,16,94],"$\\mathfrak{S}_{n}$,":[9,95],"two":[10,56],"reduced":[11],"expressions":[12],"same":[15],"element":[17,88],"$\\textsf{w}$":[18],"are":[19,58,135],"said":[20],"to":[21],"be":[22,29,46],"commutationally":[23],"equivalent":[24],"if":[25,63],"one":[26,34,124],"expression":[27],"can":[28,45],"obtained":[30],"from":[31],"other":[33],"by":[35,60,69],"applying":[36],"a":[37,52,70,102,111,128],"finite":[38],"sequence":[39],"of":[40,51,65,80,113,130,142],"commutations.":[41],"The":[42],"commutation":[43,114,131],"classes":[44,57,67,115,122,132],"seen":[47],"as":[48],"vertices":[50],"graph":[53,82,104],"$\\widehat{G}(\\textsf{w})$,":[54],"where":[55],"connected":[59],"an":[61],"edge":[62],"elements":[64,134],"those":[66],"differ":[68],"long":[71],"braid":[72],"relation.":[73],"We":[74,108],"compute":[75],"radius":[77],"and":[78,96,127],"diameter":[79],"$\\widehat{G}(\\textsf{w}_{\\bf":[83],"0})$,":[84],"longest":[87],"$\\textsf{w}_{\\bf":[89],"0}$":[90],"in":[91,136],"show":[97],"that":[98,120],"it":[99],"is":[100,121],"not":[101],"planar":[103],"$n\\geq":[106],"6$.":[107],"also":[109],"describe":[110],"family":[112],"which":[116],"contains":[117],"all":[118],"atoms,":[119],"with":[123,138],"single":[125],"element,":[126],"subfamily":[129],"whose":[133],"bijection":[137],"Young":[140],"tableaux":[141],"certain":[143],"moon-polyomino":[144],"shapes.":[145]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":2}],"updated_date":"2026-03-16T09:10:04.655348","created_date":"2025-10-10T00:00:00"}