{"id":"https://openalex.org/W2132374507","doi":"https://doi.org/10.37236/1584","title":"An Area-to-Inv Bijection Between Dyck Paths and 312-avoiding Permutations","display_name":"An Area-to-Inv Bijection Between Dyck Paths and 312-avoiding Permutations","publication_year":2001,"publication_date":"2001-12-10","ids":{"openalex":"https://openalex.org/W2132374507","doi":"https://doi.org/10.37236/1584","mag":"2132374507"},"language":"en","primary_location":{"id":"doi:10.37236/1584","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1584","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v8i1r40/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/v8i1r40/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5052875628","display_name":"Jason Bandlow","orcid":null},"institutions":[{"id":"https://openalex.org/I92446798","display_name":"Colorado State University","ror":"https://ror.org/03k1gpj17","country_code":"US","type":"education","lineage":["https://openalex.org/I92446798"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jason Bandlow","raw_affiliation_strings":["Mathematics Department Colorado State University Fort Collins, Colorado"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematics Department Colorado State University Fort Collins, Colorado","institution_ids":["https://openalex.org/I92446798"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5080635755","display_name":"Kendra Killpatrick","orcid":"https://orcid.org/0000-0002-9608-5605"},"institutions":[{"id":"https://openalex.org/I92446798","display_name":"Colorado State University","ror":"https://ror.org/03k1gpj17","country_code":"US","type":"education","lineage":["https://openalex.org/I92446798"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Kendra Killpatrick","raw_affiliation_strings":["Mathematics Department Colorado State University Fort Collins, Colorado"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematics Department Colorado State University Fort Collins, Colorado","institution_ids":["https://openalex.org/I92446798"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I92446798"],"apc_list":null,"apc_paid":null,"fwci":4.9306,"has_fulltext":true,"cited_by_count":32,"citation_normalized_percentile":{"value":0.95796178,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"8","issue":"1","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/T11428","display_name":"Advanced Mathematical Identities","score":0.9951000213623047,"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/T11269","display_name":"Algorithms and Data Compression","score":0.9807999730110168,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/catalan-number","display_name":"Catalan number","score":0.9162319898605347},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8633474111557007},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7791298627853394},{"id":"https://openalex.org/keywords/bijection","display_name":"Bijection","score":0.7733365893363953},{"id":"https://openalex.org/keywords/sigma","display_name":"Sigma","score":0.4629884958267212},{"id":"https://openalex.org/keywords/permutation","display_name":"Permutation (music)","score":0.42512571811676025},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3955162763595581}],"concepts":[{"id":"https://openalex.org/C43929256","wikidata":"https://www.wikidata.org/wiki/Q270513","display_name":"Catalan number","level":2,"score":0.9162319898605347},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8633474111557007},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7791298627853394},{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.7733365893363953},{"id":"https://openalex.org/C2778049214","wikidata":"https://www.wikidata.org/wiki/Q7512234","display_name":"Sigma","level":2,"score":0.4629884958267212},{"id":"https://openalex.org/C21308566","wikidata":"https://www.wikidata.org/wiki/Q7169365","display_name":"Permutation (music)","level":2,"score":0.42512571811676025},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3955162763595581},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","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},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.37236/1584","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1584","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v8i1r40/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:10.1.1.7.1505","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.7.1505","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_8/PDF/v8i1r40.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.9.110","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.9.110","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_8/PostScriptfiles/v8i1r40.ps","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/1584","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1584","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v8i1r40/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":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2132374507.pdf","grobid_xml":"https://content.openalex.org/works/W2132374507.grobid-xml"},"referenced_works_count":17,"referenced_works":["https://openalex.org/W1548412882","https://openalex.org/W1924676234","https://openalex.org/W2022892305","https://openalex.org/W2043605821","https://openalex.org/W2048202079","https://openalex.org/W2048655899","https://openalex.org/W2059036144","https://openalex.org/W2072036387","https://openalex.org/W2104925482","https://openalex.org/W2140141642","https://openalex.org/W2348417997","https://openalex.org/W2752061190","https://openalex.org/W2752853835","https://openalex.org/W4239240946","https://openalex.org/W4300302225","https://openalex.org/W4403296593","https://openalex.org/W6665132084"],"related_works":["https://openalex.org/W1483532407","https://openalex.org/W3109577171","https://openalex.org/W2950642020","https://openalex.org/W1721535522","https://openalex.org/W4386833582","https://openalex.org/W1596411003","https://openalex.org/W4301839635","https://openalex.org/W3004838398","https://openalex.org/W4297942051","https://openalex.org/W1980917265"],"abstract_inverted_index":{"The":[0,80,142],"symmetric":[1],"$q,t$-Catalan":[2],"polynomial":[3,10,57],"$C_n(q,t)$,":[4],"which":[5,126],"specializes":[6],"to":[7,86,135],"the":[8,42,55,75,88,97,128,136,150,164,179],"Catalan":[9,56,77,81],"$C_n(q)$":[11,58],"when":[12],"$t=1$,":[13],"was":[14],"defined":[15,59],"by":[16,60,67],"Garsia":[17,24],"and":[18,25,30,62,64,72,96,123],"Haiman":[19],"in":[20,102,155,174,176],"1994.":[21],"In":[22,113],"2000,":[23],"Haglund":[26],"described":[27],"statistics":[28],"$a(\\pi)$":[29,131],"$b(\\pi)$":[31],"on":[32,132,139],"Dyck":[33,50,94,121,133,181],"paths":[34,95,122,134],"such":[35],"that":[36,172],"$C_n(q,t)":[37],"=":[38],"\\sum_{\\pi}":[39],"q^{a(\\pi)}t^{b(\\pi)}$":[40],"where":[41],"sum":[43],"is":[44,84],"over":[45],"all":[46],"$n":[47,91],"\\times":[48,92],"n$":[49,93],"paths.":[51],"Specializing":[52,69],"$t=1$":[53,71],"gives":[54,74],"Carlitz":[61],"Riordan":[63],"further":[65],"studied":[66],"Carlitz.":[68],"both":[70],"$q=1$":[73],"usual":[76],"number":[78,82,89,98,151,165],"$C_n$.":[79],"$C_n$":[83],"known":[85],"count":[87],"of":[90,99,148,152,166,178],"$312$-avoiding":[100,124,140],"permutations":[101,125],"$S_n$,":[103],"as":[104,106,149],"well":[105],"at":[107],"least":[108],"64":[109],"other":[110],"combinatorial":[111],"objects.":[112],"this":[114],"paper,":[115],"we":[116],"define":[117],"a":[118,156,161],"bijection":[119],"between":[120],"takes":[127],"area":[129],"statistic":[130,138,144],"inversion":[137,143],"permutations.":[141],"can":[145],"be":[146],"thought":[147],"$(21)$":[153],"patterns":[154,171],"permutation":[157],"$\\sigma$.":[158],"We":[159],"give":[160],"characterization":[162],"for":[163],"$(321)$,":[167],"$(4321)$,":[168],"$\\dots$,":[169],"$(k\\cdots21)$":[170],"occur":[173],"$\\sigma$":[175],"terms":[177],"corresponding":[180],"path.":[182]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-08-26T07:47:46.906454","created_date":"2025-10-10T00:00:00"}