{"id":"https://openalex.org/W1493695725","doi":"https://doi.org/10.37236/381","title":"Records in Set Partitions","display_name":"Records in Set Partitions","publication_year":2010,"publication_date":"2010-08-09","ids":{"openalex":"https://openalex.org/W1493695725","doi":"https://doi.org/10.37236/381","mag":"1493695725"},"language":"en","primary_location":{"id":"doi:10.37236/381","is_oa":true,"landing_page_url":"https://doi.org/10.37236/381","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v17i1r109/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/v17i1r109/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5036711367","display_name":"Arnold Knopfmacher","orcid":"https://orcid.org/0000-0003-1962-043X"},"institutions":[{"id":"https://openalex.org/I192619145","display_name":"University of the Witwatersrand","ror":"https://ror.org/03rp50x72","country_code":"ZA","type":"education","lineage":["https://openalex.org/I192619145"]}],"countries":["ZA"],"is_corresponding":true,"raw_author_name":"Arnold Knopfmacher","raw_affiliation_strings":["university of the Witwatersrand"],"affiliations":[{"raw_affiliation_string":"university of the Witwatersrand","institution_ids":["https://openalex.org/I192619145"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5024416044","display_name":"Toufik Mansour","orcid":"https://orcid.org/0000-0001-8028-2391"},"institutions":[{"id":"https://openalex.org/I91203450","display_name":"University of Haifa","ror":"https://ror.org/02f009v59","country_code":"IL","type":"education","lineage":["https://openalex.org/I91203450"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Toufik Mansour","raw_affiliation_strings":["University of Haifa"],"affiliations":[{"raw_affiliation_string":"University of Haifa","institution_ids":["https://openalex.org/I91203450"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5024878725","display_name":"Stephan Wagner","orcid":null},"institutions":[{"id":"https://openalex.org/I26092322","display_name":"Stellenbosch University","ror":"https://ror.org/05bk57929","country_code":"ZA","type":"education","lineage":["https://openalex.org/I26092322"]}],"countries":["ZA"],"is_corresponding":false,"raw_author_name":"Stephan Wagner","raw_affiliation_strings":["Stellenbosch University"],"affiliations":[{"raw_affiliation_string":"Stellenbosch University","institution_ids":["https://openalex.org/I26092322"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5036711367"],"corresponding_institution_ids":["https://openalex.org/I192619145"],"apc_list":null,"apc_paid":null,"fwci":1.3672,"has_fulltext":true,"cited_by_count":16,"citation_normalized_percentile":{"value":0.79726281,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"17","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9980999827384949,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9980999827384949,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9973000288009644,"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/T11269","display_name":"Algorithms and Data Compression","score":0.9962000250816345,"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/combinatorics","display_name":"Combinatorics","score":0.8296753168106079},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8032220005989075},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.7709846496582031},{"id":"https://openalex.org/keywords/alphabet","display_name":"Alphabet","score":0.7433373928070068},{"id":"https://openalex.org/keywords/word","display_name":"Word (group theory)","score":0.5321521162986755},{"id":"https://openalex.org/keywords/generating-function","display_name":"Generating function","score":0.4284701943397522},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.36283978819847107},{"id":"https://openalex.org/keywords/linguistics","display_name":"Linguistics","score":0.051227837800979614}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8296753168106079},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8032220005989075},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.7709846496582031},{"id":"https://openalex.org/C112876837","wikidata":"https://www.wikidata.org/wiki/Q837518","display_name":"Alphabet","level":2,"score":0.7433373928070068},{"id":"https://openalex.org/C90805587","wikidata":"https://www.wikidata.org/wiki/Q10944557","display_name":"Word (group theory)","level":2,"score":0.5321521162986755},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.4284701943397522},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36283978819847107},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.051227837800979614},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.37236/381","is_oa":true,"landing_page_url":"https://doi.org/10.37236/381","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v17i1r109/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:10.1.1.176.3996","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.176.3996","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_17/PDF/v17i1r109.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/381","is_oa":true,"landing_page_url":"https://doi.org/10.37236/381","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v17i1r109/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/G2095507467","display_name":null,"funder_award_id":"70560","funder_id":"https://openalex.org/F4320320671","funder_display_name":"National Research Foundation"},{"id":"https://openalex.org/G3034753964","display_name":null,"funder_award_id":"grant","funder_id":"https://openalex.org/F4320320671","funder_display_name":"National Research Foundation"},{"id":"https://openalex.org/G3245455086","display_name":null,"funder_award_id":"2053740","funder_id":"https://openalex.org/F4320320671","funder_display_name":"National Research Foundation"}],"funders":[{"id":"https://openalex.org/F4320320671","display_name":"National Research Foundation","ror":"https://ror.org/05s0g1g46"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1493695725.pdf","grobid_xml":"https://content.openalex.org/works/W1493695725.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W186403072","https://openalex.org/W582636576","https://openalex.org/W1593478397","https://openalex.org/W1966709801","https://openalex.org/W1987029529","https://openalex.org/W1987135975","https://openalex.org/W1987392707","https://openalex.org/W1993449345","https://openalex.org/W2036825158","https://openalex.org/W2065032802","https://openalex.org/W2094678803","https://openalex.org/W2340541605","https://openalex.org/W2752061190","https://openalex.org/W2752853835","https://openalex.org/W4251062713","https://openalex.org/W4251855399"],"related_works":["https://openalex.org/W2074481494","https://openalex.org/W2808006995","https://openalex.org/W4250156143","https://openalex.org/W4247156967","https://openalex.org/W582626623","https://openalex.org/W4240940369","https://openalex.org/W4298850403","https://openalex.org/W3088566498","https://openalex.org/W2039981404","https://openalex.org/W2059650901"],"abstract_inverted_index":{"A":[0],"partition":[1,24],"of":[2,7,18,22,62,75,85,93,96,116,118,125],"$[n]=\\{1,2,\\ldots,n\\}$":[3],"is":[4,46,65],"a":[5,23,26,29,34,43,47],"decomposition":[6],"$[n]$":[8],"into":[9],"nonempty":[10],"subsets":[11],"called":[12],"blocks.":[13],"We":[14,67],"will":[15],"make":[16],"use":[17],"the":[19,60,72,91,94,97,103,110,114],"canonical":[20],"representation":[21],"as":[25,33,87,89],"word":[27,44],"over":[28],"finite":[30],"alphabet,":[31],"known":[32],"restricted":[35],"growth":[36],"function.":[37],"An":[38],"element":[39],"$a_i$":[40],"in":[41,79,122],"such":[42],"$\\pi$":[45],"strong":[48,76,119],"(weak)":[49,77,120],"record":[50,64],"if":[51],"$a_i>":[52],"a_j$":[53],"($a_i\\geq":[54],"a_j$)":[55],"for":[56,71,90,109,113],"all":[57,80,123],"$j=1,2,\\ldots,i-1$.":[58],"Furthermore,":[59],"position":[61],"this":[63],"$i$.":[66],"derive":[68],"generating":[69],"functions":[70],"total":[73],"number":[74],"records":[78,121],"words":[81],"corresponding":[82],"to":[83],"partitions":[84,124],"$[n]$,":[86],"well":[88],"sum":[92,115],"positions":[95],"records.":[98],"In":[99],"addition":[100],"we":[101],"find":[102],"asymptotic":[104],"mean":[105],"values":[106],"and":[107,112],"variances":[108],"number,":[111],"positions,":[117],"$[n]$.":[126]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"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":2},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}