{"id":"https://openalex.org/W1726889056","doi":"https://doi.org/10.37236/1264","title":"Flag-symmetric and Locally Rank-symmetric Partially Ordered Sets","display_name":"Flag-symmetric and Locally Rank-symmetric Partially Ordered Sets","publication_year":1995,"publication_date":"1995-05-26","ids":{"openalex":"https://openalex.org/W1726889056","doi":"https://doi.org/10.37236/1264","mag":"1726889056"},"language":"en","primary_location":{"id":"doi:10.37236/1264","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1264","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v3i2r6/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":false,"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":null,"license_id":null,"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":"bronze","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v3i2r6/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5079674634","display_name":"Richard P. Stanley","orcid":null},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Richard P. Stanley","raw_affiliation_strings":["Massachusetts Institute of Technology"],"affiliations":[{"raw_affiliation_string":"Massachusetts Institute of Technology","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5079674634"],"corresponding_institution_ids":["https://openalex.org/I63966007"],"apc_list":null,"apc_paid":null,"fwci":2.2819,"has_fulltext":true,"cited_by_count":37,"citation_normalized_percentile":{"value":0.83407276,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":95},"biblio":{"volume":"3","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":0.9998999834060669,"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":0.9998999834060669,"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/T10287","display_name":"Algebraic structures and combinatorial models","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8340761661529541},{"id":"https://openalex.org/keywords/symmetric-function","display_name":"Symmetric function","score":0.8134629726409912},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7349873781204224},{"id":"https://openalex.org/keywords/ring-of-symmetric-functions","display_name":"Ring of symmetric functions","score":0.6556107997894287},{"id":"https://openalex.org/keywords/symmetric-group","display_name":"Symmetric group","score":0.6404911279678345},{"id":"https://openalex.org/keywords/lexicographical-order","display_name":"Lexicographical order","score":0.5861303210258484},{"id":"https://openalex.org/keywords/schur-polynomial","display_name":"Schur polynomial","score":0.5850374698638916},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.5646539330482483},{"id":"https://openalex.org/keywords/symmetric-polynomial","display_name":"Symmetric polynomial","score":0.5466788411140442},{"id":"https://openalex.org/keywords/partially-ordered-set","display_name":"Partially ordered set","score":0.5195370316505432},{"id":"https://openalex.org/keywords/schur-algebra","display_name":"Schur algebra","score":0.5120332837104797},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.5046542882919312},{"id":"https://openalex.org/keywords/stanley-symmetric-function","display_name":"Stanley symmetric function","score":0.4986233711242676},{"id":"https://openalex.org/keywords/flag","display_name":"Flag (linear algebra)","score":0.4940686523914337},{"id":"https://openalex.org/keywords/bruhat-order","display_name":"Bruhat order","score":0.47376325726509094},{"id":"https://openalex.org/keywords/power-sum-symmetric-polynomial","display_name":"Power sum symmetric polynomial","score":0.4618028998374939},{"id":"https://openalex.org/keywords/elementary-symmetric-polynomial","display_name":"Elementary symmetric polynomial","score":0.4443321228027344},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.41383060812950134},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.23666921257972717},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.19170942902565002},{"id":"https://openalex.org/keywords/macdonald-polynomials","display_name":"Macdonald polynomials","score":0.18136781454086304},{"id":"https://openalex.org/keywords/coxeter-group","display_name":"Coxeter group","score":0.15949615836143494},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.12630480527877808},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.10944297909736633},{"id":"https://openalex.org/keywords/difference-polynomials","display_name":"Difference polynomials","score":0.09772899746894836}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8340761661529541},{"id":"https://openalex.org/C3746008","wikidata":"https://www.wikidata.org/wiki/Q981351","display_name":"Symmetric function","level":2,"score":0.8134629726409912},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7349873781204224},{"id":"https://openalex.org/C156903151","wikidata":"https://www.wikidata.org/wiki/Q17102788","display_name":"Ring of symmetric functions","level":4,"score":0.6556107997894287},{"id":"https://openalex.org/C128622974","wikidata":"https://www.wikidata.org/wiki/Q849512","display_name":"Symmetric group","level":2,"score":0.6404911279678345},{"id":"https://openalex.org/C159254197","wikidata":"https://www.wikidata.org/wiki/Q1144915","display_name":"Lexicographical order","level":2,"score":0.5861303210258484},{"id":"https://openalex.org/C146922319","wikidata":"https://www.wikidata.org/wiki/Q4298935","display_name":"Schur polynomial","level":5,"score":0.5850374698638916},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5646539330482483},{"id":"https://openalex.org/C52153879","wikidata":"https://www.wikidata.org/wiki/Q930499","display_name":"Symmetric polynomial","level":4,"score":0.5466788411140442},{"id":"https://openalex.org/C180645754","wikidata":"https://www.wikidata.org/wiki/Q474715","display_name":"Partially ordered set","level":2,"score":0.5195370316505432},{"id":"https://openalex.org/C43562168","wikidata":"https://www.wikidata.org/wiki/Q7433031","display_name":"Schur algebra","level":5,"score":0.5120332837104797},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.5046542882919312},{"id":"https://openalex.org/C78136718","wikidata":"https://www.wikidata.org/wiki/Q7600073","display_name":"Stanley symmetric function","level":5,"score":0.4986233711242676},{"id":"https://openalex.org/C2776730729","wikidata":"https://www.wikidata.org/wiki/Q1392130","display_name":"Flag (linear algebra)","level":3,"score":0.4940686523914337},{"id":"https://openalex.org/C177274530","wikidata":"https://www.wikidata.org/wiki/Q4978697","display_name":"Bruhat order","level":3,"score":0.47376325726509094},{"id":"https://openalex.org/C129010115","wikidata":"https://www.wikidata.org/wiki/Q1298683","display_name":"Power sum symmetric polynomial","level":5,"score":0.4618028998374939},{"id":"https://openalex.org/C64338288","wikidata":"https://www.wikidata.org/wiki/Q2235316","display_name":"Elementary symmetric polynomial","level":4,"score":0.4443321228027344},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.41383060812950134},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.23666921257972717},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.19170942902565002},{"id":"https://openalex.org/C171686973","wikidata":"https://www.wikidata.org/wiki/Q17098676","display_name":"Macdonald polynomials","level":4,"score":0.18136781454086304},{"id":"https://openalex.org/C143669375","wikidata":"https://www.wikidata.org/wiki/Q7874246","display_name":"Coxeter group","level":2,"score":0.15949615836143494},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.12630480527877808},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.10944297909736633},{"id":"https://openalex.org/C86607863","wikidata":"https://www.wikidata.org/wiki/Q5275279","display_name":"Difference polynomials","level":3,"score":0.09772899746894836},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C78540521","wikidata":"https://www.wikidata.org/wiki/Q1498246","display_name":"Gegenbauer polynomials","level":4,"score":0.0},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.37236/1264","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1264","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v3i2r6/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":false,"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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.29.9130","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.9130","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ftp://ftp.maths.tcd.ie/pub/EMIS/journals/EJC/Volume_3/PostScriptfiles/v3i2r6.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.34.5393","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.5393","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.ras.ru/EMIS/journals/EJC/Volume_3/PDFFiles/v3i2r6.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.67.9406","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.67.9406","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://cs.anu.edu.au/publications/eljc/Volume_3/PostScriptfiles/v3i2r6.ps","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/1264","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1264","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v3i2r6/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":false,"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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1726889056.pdf"},"referenced_works_count":32,"referenced_works":["https://openalex.org/W1527908630","https://openalex.org/W1557033434","https://openalex.org/W1565240535","https://openalex.org/W1599997798","https://openalex.org/W1963521086","https://openalex.org/W1975549371","https://openalex.org/W1982899022","https://openalex.org/W1985177626","https://openalex.org/W2003891445","https://openalex.org/W2009190506","https://openalex.org/W2013502334","https://openalex.org/W2015050395","https://openalex.org/W2054802601","https://openalex.org/W2062537955","https://openalex.org/W2072173760","https://openalex.org/W2077309624","https://openalex.org/W2082205190","https://openalex.org/W2083882790","https://openalex.org/W2134984950","https://openalex.org/W2165125983","https://openalex.org/W2405154940","https://openalex.org/W2486012592","https://openalex.org/W2914471578","https://openalex.org/W2914659449","https://openalex.org/W4213139606","https://openalex.org/W4229504661","https://openalex.org/W4237012419","https://openalex.org/W4238933498","https://openalex.org/W4243425154","https://openalex.org/W4254449038","https://openalex.org/W4300302225","https://openalex.org/W6697549010"],"related_works":["https://openalex.org/W2388754783","https://openalex.org/W3045701420","https://openalex.org/W2901510162","https://openalex.org/W2949691349","https://openalex.org/W1560962904","https://openalex.org/W3013709019","https://openalex.org/W188141326","https://openalex.org/W2401068015","https://openalex.org/W4211222368","https://openalex.org/W1982680725"],"abstract_inverted_index":{"For":[0,85],"every":[1],"finite":[2],"graded":[3],"poset":[4],"$P$":[5],"with":[6,150],"$\\hat{0}$":[7],"and":[8,158],"$\\hat{1}$":[9],"we":[10,70],"associate":[11],"a":[12,40,59,86,112],"certain":[13],"formal":[14],"power":[15,106],"series":[16],"$F_P(x)":[17],"=":[18],"F_P(x_1,x_2,\\dots)$":[19],"which":[20],"encodes":[21],"the":[22,43,52,82,93,104,126,133,151,155],"flag":[23,26],"$f$-vector":[24],"(or":[25],"$h$-vector)":[27],"of":[28,42,46,62,76,88,100,116,121,128,154],"$P$.":[29,47],"A":[30],"relative":[31,134,145],"version":[32],"$F_{P/\\Gamma}$":[33,57,67],"is":[34,39,58,68],"also":[35,163],"defined,":[36],"where":[37,54],"$\\Gamma$":[38],"subcomplex":[41],"order":[44,137],"complex":[45],"We":[48,124],"are":[49,97,144,162],"interested":[50],"in":[51,74,110,119,136],"situation":[53],"$F_P$":[55,65],"or":[56,66],"symmetric":[60,69,78,122,156],"function":[61,79,95],"$x_1,x_2,\\dots$.":[63],"When":[64],"consider":[71],"its":[72,159],"expansion":[73],"terms":[75,120],"various":[77],"bases,":[80],"especially":[81],"Schur":[83,94],"functions.":[84,123],"class":[87],"lattices":[89,92],"called":[90],"$q$-primary":[91],"coefficients":[96],"just":[98],"values":[99],"Kostka":[101,117],"polynomials":[102,118],"at":[103],"prime":[105],"$q$,":[107],"thus":[108],"giving":[109],"effect":[111],"simple":[113],"new":[114],"definition":[115],"extend":[125],"theory":[127,153],"lexicographically":[129],"shellable":[130],"posets":[131],"to":[132,138],"case":[135],"show":[139],"that":[140],"some":[141],"examples":[142],"$(P,\\Gamma)$":[143],"Cohen-Macaulay":[146],"complexes.":[147],"Some":[148],"connections":[149],"representation":[152],"group":[157],"Hecke":[160],"algebra":[161],"discussed.":[164]},"counts_by_year":[{"year":2021,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}
