{"id":"https://openalex.org/W1519715486","doi":"https://doi.org/10.37236/1677","title":"$321$-Polygon-Avoiding Permutations and Chebyshev Polynomials","display_name":"$321$-Polygon-Avoiding Permutations and Chebyshev Polynomials","publication_year":2003,"publication_date":"2003-01-22","ids":{"openalex":"https://openalex.org/W1519715486","doi":"https://doi.org/10.37236/1677","mag":"1519715486"},"language":"en","primary_location":{"id":"doi:10.37236/1677","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1677","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v9i2r5/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/v9i2r5/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5024416044","display_name":"Toufik Mansour","orcid":"https://orcid.org/0000-0001-8028-2391"},"institutions":[{"id":"https://openalex.org/I4210142254","display_name":"Laboratoire Bordelais de Recherche en Informatique","ror":"https://ror.org/03adqg323","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I15057530","https://openalex.org/I4210142254","https://openalex.org/I4210159245","https://openalex.org/I4210160189"]},{"id":"https://openalex.org/I15057530","display_name":"Universit\u00e9 de Bordeaux","ror":"https://ror.org/057qpr032","country_code":"FR","type":"education","lineage":["https://openalex.org/I15057530"]},{"id":"https://openalex.org/I4210166017","display_name":"Institut de Math\u00e9matiques de Bordeaux","ror":"https://ror.org/05m3r1b84","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I15057530","https://openalex.org/I4210141950","https://openalex.org/I4210160189","https://openalex.org/I4210166017"]},{"id":"https://openalex.org/I4210160189","display_name":"Institut Polytechnique de Bordeaux","ror":"https://ror.org/054qv7y42","country_code":"FR","type":"education","lineage":["https://openalex.org/I4210160189"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Toufik Mansour","raw_affiliation_strings":["LaBRI, Universit Bordeaux I, 351 cours de la Libration, 33405 Talence Cedex, France","LaBRI, Universit\u00e9 Bordeaux I, 351 cours de la Lib\u00e9ration, 33405 Talence Cedex, France"],"affiliations":[{"raw_affiliation_string":"LaBRI, Universit Bordeaux I, 351 cours de la Libration, 33405 Talence Cedex, France","institution_ids":["https://openalex.org/I4210166017","https://openalex.org/I15057530","https://openalex.org/I4210142254"]},{"raw_affiliation_string":"LaBRI, Universit\u00e9 Bordeaux I, 351 cours de la Lib\u00e9ration, 33405 Talence Cedex, France","institution_ids":["https://openalex.org/I15057530","https://openalex.org/I4210142254","https://openalex.org/I4210160189"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5068759323","display_name":"Zvezdelina Stankova","orcid":null},"institutions":[{"id":"https://openalex.org/I91481961","display_name":"Mills College","ror":"https://ror.org/01ynb8e46","country_code":"US","type":"education","lineage":["https://openalex.org/I12912129","https://openalex.org/I91481961"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zvezdelina Stankova","raw_affiliation_strings":["Mills College, Oakland, CA"],"affiliations":[{"raw_affiliation_string":"Mills College, Oakland, CA","institution_ids":["https://openalex.org/I91481961"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5024416044"],"corresponding_institution_ids":["https://openalex.org/I15057530","https://openalex.org/I4210142254","https://openalex.org/I4210160189","https://openalex.org/I4210166017"],"apc_list":null,"apc_paid":null,"fwci":3.8596,"has_fulltext":true,"cited_by_count":16,"citation_normalized_percentile":{"value":0.92713568,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"9","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":0.9997000098228455,"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":0.9997000098228455,"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/T11428","display_name":"Advanced Mathematical Identities","score":0.9932000041007996,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T13506","display_name":"Botanical Research and Chemistry","score":0.9876000285148621,"subfield":{"id":"https://openalex.org/subfields/1105","display_name":"Ecology, Evolution, Behavior and Systematics"},"field":{"id":"https://openalex.org/fields/11","display_name":"Agricultural and Biological Sciences"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8338422775268555},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7941240072250366},{"id":"https://openalex.org/keywords/generating-function","display_name":"Generating function","score":0.7205739617347717},{"id":"https://openalex.org/keywords/permutation","display_name":"Permutation (music)","score":0.7144219875335693},{"id":"https://openalex.org/keywords/chebyshev-polynomials","display_name":"Chebyshev polynomials","score":0.6716001629829407},{"id":"https://openalex.org/keywords/enumeration","display_name":"Enumeration","score":0.5399027466773987},{"id":"https://openalex.org/keywords/polygon","display_name":"Polygon (computer graphics)","score":0.5204672813415527},{"id":"https://openalex.org/keywords/symmetric-group","display_name":"Symmetric group","score":0.4727643132209778},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.43586188554763794},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.4352670907974243},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.4237675070762634},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3342565894126892},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.058804064989089966},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.04785192012786865}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8338422775268555},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7941240072250366},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.7205739617347717},{"id":"https://openalex.org/C21308566","wikidata":"https://www.wikidata.org/wiki/Q7169365","display_name":"Permutation (music)","level":2,"score":0.7144219875335693},{"id":"https://openalex.org/C129785596","wikidata":"https://www.wikidata.org/wiki/Q619511","display_name":"Chebyshev polynomials","level":2,"score":0.6716001629829407},{"id":"https://openalex.org/C156340839","wikidata":"https://www.wikidata.org/wiki/Q2704791","display_name":"Enumeration","level":2,"score":0.5399027466773987},{"id":"https://openalex.org/C190694206","wikidata":"https://www.wikidata.org/wiki/Q3276654","display_name":"Polygon (computer graphics)","level":3,"score":0.5204672813415527},{"id":"https://openalex.org/C128622974","wikidata":"https://www.wikidata.org/wiki/Q849512","display_name":"Symmetric group","level":2,"score":0.4727643132209778},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.43586188554763794},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.4352670907974243},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.4237675070762634},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3342565894126892},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.058804064989089966},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.04785192012786865},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C126042441","wikidata":"https://www.wikidata.org/wiki/Q1324888","display_name":"Frame (networking)","level":2,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.37236/1677","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1677","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v9i2r5/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:10.1.1.239.7840","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.239.7840","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/math/0207192v1.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.3.652","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.652","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.combinatorics.org/Volume_9/PDF/v9i2r5.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.9.3982","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.9.3982","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.combinatorics.org/Volume_9/PostScriptfiles/v9i2r5.ps","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/1677","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1677","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v9i2r5/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/F4320321523","display_name":"Chalmers Tekniska H\u00f6gskola","ror":"https://ror.org/040wg7k59"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1519715486.pdf","grobid_xml":"https://content.openalex.org/works/W1519715486.grobid-xml"},"referenced_works_count":23,"referenced_works":["https://openalex.org/W129807151","https://openalex.org/W209550864","https://openalex.org/W1500943703","https://openalex.org/W1538027092","https://openalex.org/W1678540464","https://openalex.org/W1706877037","https://openalex.org/W1968120658","https://openalex.org/W2003628938","https://openalex.org/W2047160913","https://openalex.org/W2048655899","https://openalex.org/W2064727561","https://openalex.org/W2074370694","https://openalex.org/W2104925482","https://openalex.org/W2129036921","https://openalex.org/W2171905396","https://openalex.org/W2752061190","https://openalex.org/W2752853835","https://openalex.org/W2950382558","https://openalex.org/W2952402095","https://openalex.org/W2952550150","https://openalex.org/W4212838771","https://openalex.org/W4235551663","https://openalex.org/W6682324371"],"related_works":["https://openalex.org/W1982980084","https://openalex.org/W2024532720","https://openalex.org/W2150561647","https://openalex.org/W2951064730","https://openalex.org/W4300993449","https://openalex.org/W2003628938","https://openalex.org/W2951624858","https://openalex.org/W2110201181","https://openalex.org/W1995202720","https://openalex.org/W2064421824"],"abstract_inverted_index":{"A":[0],"$321$-$k$-gon-avoiding":[1,96],"permutation":[2],"$\\pi$":[3],"avoids":[4],"$321$":[5],"and":[6,22,26,42,54,57],"the":[7,35,43,73,89,93,110],"following":[8],"four":[9],"patterns:":[10],"$$k(k+2)(k+3)\\cdots(2k-1)1(2k)23\\cdots(k-1)(k+1),$$":[11],"$$k(k+2)(k+3)\\cdots(2k-1)(2k)12\\cdots(k-1)(k+1),$$":[12],"$$(k+1)(k+2)(k+3)\\cdots(2k-1)1(2k)23\\cdots":[13],"k,$$":[14],"$$(k+1)(k+2)(k+3)\\cdots(2k-1)(2k)123\\cdots":[15],"k.$$":[16],"The":[17,101],"$321$-$4$-gon-avoiding":[18],"permutations":[19,97],"were":[20],"introduced":[21],"studied":[23],"by":[24,83],"Billey":[25],"Warrington":[27],"[BW]":[28],"as":[29],"a":[30],"class":[31],"of":[32,34,46,66,95,109],"elements":[33],"symmetric":[36],"group":[37],"whose":[38,47],"Kazhdan-Lusztig,":[39],"Poincar\u00e9":[40],"polynomials,":[41],"singular":[44],"loci":[45],"Schubert":[48],"varieties":[49],"have":[50],"fairly":[51],"simple":[52],"formulas":[53],"descriptions.":[55],"Stankova":[56],"West":[58],"[SW]":[59],"gave":[60],"an":[61,85],"exact":[62],"enumeration":[63],"in":[64],"terms":[65],"linear":[67],"recurrences":[68],"with":[69],"constant":[70],"coefficients":[71],"for":[72,88,92],"cases":[74],"$k=2,3,4$.":[75],"In":[76],"this":[77],"paper,":[78],"we":[79],"extend":[80],"these":[81],"results":[82],"finding":[84],"explicit":[86],"expression":[87],"generating":[90,102],"function":[91,103],"number":[94],"on":[98],"$n$":[99],"letters.":[100],"is":[104],"expressed":[105],"via":[106],"Chebyshev":[107],"polynomials":[108],"second":[111],"kind.":[112]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":2}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}