{"id":"https://openalex.org/W1583030682","doi":"https://doi.org/10.37236/1922","title":"Wilf Classes of Pairs of Permutations of Length 4","display_name":"Wilf Classes of Pairs of Permutations of Length 4","publication_year":2005,"publication_date":"2005-05-26","ids":{"openalex":"https://openalex.org/W1583030682","doi":"https://doi.org/10.37236/1922","mag":"1583030682"},"language":"en","primary_location":{"id":"doi:10.37236/1922","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1922","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v12i1r25/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":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":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v12i1r25/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5064199361","display_name":"Ian Le","orcid":null},"institutions":[{"id":"https://openalex.org/I20089843","display_name":"Princeton University","ror":"https://ror.org/00hx57361","country_code":"US","type":"education","lineage":["https://openalex.org/I20089843"]},{"id":"https://openalex.org/I2801851002","display_name":"Harvard University Press","ror":"https://ror.org/006v7bf86","country_code":"US","type":"other","lineage":["https://openalex.org/I136199984","https://openalex.org/I2801851002"]},{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Ian Le","raw_affiliation_strings":["Mathematics Department Harvard University 1 Oxford St., Cambridge, MA 02138","Princeton Jct., NJ 08550","Harvard University"],"affiliations":[{"raw_affiliation_string":"Mathematics Department Harvard University 1 Oxford St., Cambridge, MA 02138","institution_ids":["https://openalex.org/I136199984"]},{"raw_affiliation_string":"Princeton Jct., NJ 08550","institution_ids":["https://openalex.org/I20089843"]},{"raw_affiliation_string":"Harvard University","institution_ids":["https://openalex.org/I2801851002"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5064199361"],"corresponding_institution_ids":["https://openalex.org/I136199984","https://openalex.org/I20089843","https://openalex.org/I2801851002"],"apc_list":null,"apc_paid":null,"fwci":1.3136,"has_fulltext":true,"cited_by_count":27,"citation_normalized_percentile":{"value":0.74472858,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"12","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":0.9997000098228455,"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/T11130","display_name":"Coding theory and cryptography","score":0.9914000034332275,"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"}},{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":0.9907000064849854,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/bijection-injection-and-surjection","display_name":"Bijection, injection and surjection","score":0.9231171011924744},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8444807529449463},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8112983107566833},{"id":"https://openalex.org/keywords/subsequence","display_name":"Subsequence","score":0.7544782161712646},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.5120487809181213},{"id":"https://openalex.org/keywords/block","display_name":"Block (permutation group theory)","score":0.49750545620918274},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.44953256845474243},{"id":"https://openalex.org/keywords/longest-increasing-subsequence","display_name":"Longest increasing subsequence","score":0.4439452886581421},{"id":"https://openalex.org/keywords/generating-function","display_name":"Generating function","score":0.423037588596344},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.41225147247314453},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.34741538763046265},{"id":"https://openalex.org/keywords/bijection","display_name":"Bijection","score":0.17938575148582458}],"concepts":[{"id":"https://openalex.org/C48659774","wikidata":"https://www.wikidata.org/wiki/Q4907197","display_name":"Bijection, injection and surjection","level":3,"score":0.9231171011924744},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8444807529449463},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8112983107566833},{"id":"https://openalex.org/C137877099","wikidata":"https://www.wikidata.org/wiki/Q1332977","display_name":"Subsequence","level":3,"score":0.7544782161712646},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5120487809181213},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.49750545620918274},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.44953256845474243},{"id":"https://openalex.org/C170006305","wikidata":"https://www.wikidata.org/wiki/Q4183855","display_name":"Longest increasing subsequence","level":4,"score":0.4439452886581421},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.423037588596344},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.41225147247314453},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.34741538763046265},{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.17938575148582458},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","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/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.37236/1922","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1922","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v12i1r25/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":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:10.1.1.214.9971","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.214.9971","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_12/PDF/v12i1r25.pdf","raw_type":"text"},{"id":"pmh:oai:openresearch-repository.anu.edu.au:1885/733764803","is_oa":false,"landing_page_url":"https://hdl.handle.net/1885/733764803","pdf_url":null,"source":{"id":"https://openalex.org/S4306402539","display_name":"ANU Open Research (Australian National University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I118347636","host_organization_name":"Australian National University","host_organization_lineage":["https://openalex.org/I118347636"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Electronic Journal of Combinatorics","raw_type":"Journal article"}],"best_oa_location":{"id":"doi:10.37236/1922","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1922","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v12i1r25/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":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":[{"id":"https://openalex.org/G652434006","display_name":"REU Site: An Undergraduate Research Experience Program in Discrete Mathematics","funder_award_id":"0137611","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320309636","display_name":"University of Minnesota","ror":"https://ror.org/03grvy078"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1583030682.pdf","grobid_xml":"https://content.openalex.org/works/W1583030682.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W1706877037","https://openalex.org/W2024633196","https://openalex.org/W4212838771","https://openalex.org/W6808399187"],"related_works":["https://openalex.org/W3122601395","https://openalex.org/W2969919693","https://openalex.org/W2125120904","https://openalex.org/W4225868058","https://openalex.org/W2035123411","https://openalex.org/W2112772040","https://openalex.org/W2131191265","https://openalex.org/W4389489786","https://openalex.org/W1540010783","https://openalex.org/W2075403678"],"abstract_inverted_index":{"$S_n(\\pi_1,\\pi_2,\\dots,":[0],"\\pi_r)$":[1],"denotes":[2],"the":[3,15,22,37,56,59,66],"set":[4],"of":[5,7,21,39,44,46,61],"permutations":[6,45],"length":[8,47],"$n$":[9],"that":[10,29],"have":[11],"no":[12],"subsequence":[13],"with":[14],"same":[16],"order":[17],"relations":[18],"as":[19],"any":[20],"$\\pi_i$.":[23],"In":[24,49],"this":[25],"paper":[26],"we":[27,52,68],"show":[28],"$|S_n(1342,2143)|=|S_n(3142,2341)|$":[30],"and":[31,64],"$|S_n(1342,3124)|=|S_n(1243,2134)|$.":[32],"These":[33],"two":[34],"facts":[35],"complete":[36],"classification":[38],"Wilf-equivalence":[40],"classes":[41],"for":[42,73],"pairs":[43],"four.":[48],"both":[50],"instances":[51],"exhibit":[53],"bijections":[54],"between":[55],"sets":[57],"using":[58],"idea":[60],"a":[62,70],"\"block\",":[63],"in":[65],"former":[67],"find":[69],"generating":[71],"function":[72],"$|S_n(1342,2143)|$.":[74]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":3},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":3}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}
