{"id":"https://openalex.org/W2148523660","doi":"https://doi.org/10.37236/1156","title":"BG-Ranks and 2-Cores","display_name":"BG-Ranks and 2-Cores","publication_year":2006,"publication_date":"2006-11-06","ids":{"openalex":"https://openalex.org/W2148523660","doi":"https://doi.org/10.37236/1156","mag":"2148523660"},"language":"en","primary_location":{"id":"doi:10.37236/1156","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1156","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v13i1n18/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/v13i1n18/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5060654252","display_name":"William Y. C. Chen","orcid":null},"institutions":[{"id":"https://openalex.org/I205237279","display_name":"Nankai University","ror":"https://ror.org/01y1kjr75","country_code":"CN","type":"education","lineage":["https://openalex.org/I205237279"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"William Y. C. Chen","raw_affiliation_strings":["Center for Combinatorics, LPMC Nankai University, Tianjin 300071, P. R. China"],"affiliations":[{"raw_affiliation_string":"Center for Combinatorics, LPMC Nankai University, Tianjin 300071, P. R. China","institution_ids":["https://openalex.org/I205237279"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5040502431","display_name":"Kathy Q. Ji","orcid":"https://orcid.org/0000-0001-7658-9462"},"institutions":[{"id":"https://openalex.org/I205237279","display_name":"Nankai University","ror":"https://ror.org/01y1kjr75","country_code":"CN","type":"education","lineage":["https://openalex.org/I205237279"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Kathy Q. Ji","raw_affiliation_strings":["Center for Combinatorics, LPMC Nankai University, Tianjin 300071, P. R. China"],"affiliations":[{"raw_affiliation_string":"Center for Combinatorics, LPMC Nankai University, Tianjin 300071, P. R. China","institution_ids":["https://openalex.org/I205237279"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5112203336","display_name":"Herbert S. Wilf","orcid":null},"institutions":[{"id":"https://openalex.org/I79576946","display_name":"University of Pennsylvania","ror":"https://ror.org/00b30xv10","country_code":"US","type":"education","lineage":["https://openalex.org/I79576946"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Herbert S. Wilf","raw_affiliation_strings":["Department of Mathematics University of Pennsylvania Philadelphia, PA 19104, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics University of Pennsylvania Philadelphia, PA 19104, USA","institution_ids":["https://openalex.org/I79576946"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5060654252"],"corresponding_institution_ids":["https://openalex.org/I205237279"],"apc_list":null,"apc_paid":null,"fwci":0.5758,"has_fulltext":true,"cited_by_count":7,"citation_normalized_percentile":{"value":0.68494858,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"13","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":1.0,"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"}},"topics":[{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":1.0,"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/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/T11166","display_name":"Analytic Number Theory Research","score":0.9991000294685364,"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/congruence-relation","display_name":"Congruence relation","score":0.9489516019821167},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8741471767425537},{"id":"https://openalex.org/keywords/bijection","display_name":"Bijection","score":0.8630426526069641},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7719024419784546},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.7705354690551758},{"id":"https://openalex.org/keywords/generating-function","display_name":"Generating function","score":0.5648756623268127},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.5325457453727722},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3712356984615326},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.039209604263305664}],"concepts":[{"id":"https://openalex.org/C9973445","wikidata":"https://www.wikidata.org/wiki/Q8349849","display_name":"Congruence relation","level":2,"score":0.9489516019821167},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8741471767425537},{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.8630426526069641},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7719024419784546},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.7705354690551758},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.5648756623268127},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.5325457453727722},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3712356984615326},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.039209604263305664}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/1156","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1156","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v13i1n18/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"}],"best_oa_location":{"id":"doi:10.37236/1156","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1156","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v13i1n18/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":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2148523660.pdf","grobid_xml":"https://content.openalex.org/works/W2148523660.grobid-xml"},"referenced_works_count":17,"referenced_works":["https://openalex.org/W1503154552","https://openalex.org/W1514324498","https://openalex.org/W1664546881","https://openalex.org/W1693715085","https://openalex.org/W1964902440","https://openalex.org/W1994284482","https://openalex.org/W1999607671","https://openalex.org/W2076597612","https://openalex.org/W2083163566","https://openalex.org/W2090865800","https://openalex.org/W2092982296","https://openalex.org/W2134984950","https://openalex.org/W2159552098","https://openalex.org/W2488278335","https://openalex.org/W2551331446","https://openalex.org/W3175367423","https://openalex.org/W4388288416"],"related_works":["https://openalex.org/W4206673079","https://openalex.org/W2355635068","https://openalex.org/W4309131384","https://openalex.org/W2048655899","https://openalex.org/W2950049363","https://openalex.org/W4386833582","https://openalex.org/W2292001013","https://openalex.org/W2982742402","https://openalex.org/W2794199602","https://openalex.org/W4388583127"],"abstract_inverted_index":{"We":[0],"find":[1,38],"the":[2,16],"number":[3,17,24],"of":[4,6,14,18,20,25,54,58],"partitions":[5,21],"$n$":[7],"whose":[8,22],"BG-rank":[9],"is":[10,27],"$j$,":[11],"in":[12],"terms":[13],"$pp(n)$,":[15,43],"pairs":[19],"total":[23],"cells":[26],"$n$,":[28],"giving":[29],"both":[30],"bijective":[31],"and":[32,44,67],"generating":[33],"function":[34],"proofs.":[35],"Next":[36],"we":[37,46],"congruences":[39,59],"mod":[40],"5":[41],"for":[42,60],"then":[45],"use":[47],"these":[48],"to":[49],"give":[50],"a":[51,55],"new":[52],"proof":[53],"refined":[56],"system":[57],"$p(n)$":[61],"that":[62],"was":[63],"found":[64],"by":[65],"Berkovich":[66],"Garvan.":[68]},"counts_by_year":[{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}