{"id":"https://openalex.org/W4253398483","doi":"https://doi.org/10.37236/1370","title":"Durfee Polynomials","display_name":"Durfee Polynomials","publication_year":1998,"publication_date":"1998-06-10","ids":{"openalex":"https://openalex.org/W4253398483","doi":"https://doi.org/10.37236/1370"},"language":"en","primary_location":{"id":"doi:10.37236/1370","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1370","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v5i1r32/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/v5i1r32/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5071772779","display_name":"E. Rodney Canfield","orcid":null},"institutions":[{"id":"https://openalex.org/I102197404","display_name":"Universit\u00e9 Paris-Sud","ror":"https://ror.org/028rypz17","country_code":"FR","type":"education","lineage":["https://openalex.org/I102197404"]},{"id":"https://openalex.org/I4210144804","display_name":"Laboratoire de Recherche en Informatique","ror":"https://ror.org/04e3ktk27","country_code":"FR","type":"facility","lineage":["https://openalex.org/I102197404","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I4210144804","https://openalex.org/I4210159245"]},{"id":"https://openalex.org/I165733156","display_name":"University of Georgia","ror":"https://ror.org/00te3t702","country_code":"US","type":"education","lineage":["https://openalex.org/I165733156"]},{"id":"https://openalex.org/I137902535","display_name":"North Carolina State University","ror":"https://ror.org/04tj63d06","country_code":"US","type":"education","lineage":["https://openalex.org/I137902535"]}],"countries":["FR","US"],"is_corresponding":true,"raw_author_name":"E. Rodney Canfield","raw_affiliation_strings":["Department of Computer Science North Carolina State University Raleigh, NC 27695-8206, USA","Department of Computer Science University of Georgia Athens, GA 30602, USA","Laboratoire de Recherche en Informatique B\u00e2t 490, Universit\u00e9 Paris-Sud 91405 Orsay, FRANCE"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science North Carolina State University Raleigh, NC 27695-8206, USA","institution_ids":["https://openalex.org/I137902535"]},{"raw_affiliation_string":"Department of Computer Science University of Georgia Athens, GA 30602, USA","institution_ids":["https://openalex.org/I165733156"]},{"raw_affiliation_string":"Laboratoire de Recherche en Informatique B\u00e2t 490, Universit\u00e9 Paris-Sud 91405 Orsay, FRANCE","institution_ids":["https://openalex.org/I4210144804","https://openalex.org/I102197404"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5038635373","display_name":"Sylvie Corteel","orcid":"https://orcid.org/0000-0003-4669-4653"},"institutions":[{"id":"https://openalex.org/I165733156","display_name":"University of Georgia","ror":"https://ror.org/00te3t702","country_code":"US","type":"education","lineage":["https://openalex.org/I165733156"]},{"id":"https://openalex.org/I102197404","display_name":"Universit\u00e9 Paris-Sud","ror":"https://ror.org/028rypz17","country_code":"FR","type":"education","lineage":["https://openalex.org/I102197404"]},{"id":"https://openalex.org/I137902535","display_name":"North Carolina State University","ror":"https://ror.org/04tj63d06","country_code":"US","type":"education","lineage":["https://openalex.org/I137902535"]},{"id":"https://openalex.org/I4210144804","display_name":"Laboratoire de Recherche en Informatique","ror":"https://ror.org/04e3ktk27","country_code":"FR","type":"facility","lineage":["https://openalex.org/I102197404","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I4210144804","https://openalex.org/I4210159245"]}],"countries":["FR","US"],"is_corresponding":false,"raw_author_name":"Sylvie Corteel","raw_affiliation_strings":["Department of Computer Science North Carolina State University Raleigh, NC 27695-8206, USA","Laboratoire de Recherche en Informatique B\u00e2t 490, Universit\u00e9 Paris-Sud 91405 Orsay, FRANCE","Department of Computer Science University of Georgia Athens, GA 30602, USA"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science North Carolina State University Raleigh, NC 27695-8206, USA","institution_ids":["https://openalex.org/I137902535"]},{"raw_affiliation_string":"Laboratoire de Recherche en Informatique B\u00e2t 490, Universit\u00e9 Paris-Sud 91405 Orsay, FRANCE","institution_ids":["https://openalex.org/I4210144804","https://openalex.org/I102197404"]},{"raw_affiliation_string":"Department of Computer Science University of Georgia Athens, GA 30602, USA","institution_ids":["https://openalex.org/I165733156"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5029085327","display_name":"Carla D. Savage","orcid":null},"institutions":[{"id":"https://openalex.org/I4210144804","display_name":"Laboratoire de Recherche en Informatique","ror":"https://ror.org/04e3ktk27","country_code":"FR","type":"facility","lineage":["https://openalex.org/I102197404","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I4210144804","https://openalex.org/I4210159245"]},{"id":"https://openalex.org/I102197404","display_name":"Universit\u00e9 Paris-Sud","ror":"https://ror.org/028rypz17","country_code":"FR","type":"education","lineage":["https://openalex.org/I102197404"]},{"id":"https://openalex.org/I165733156","display_name":"University of Georgia","ror":"https://ror.org/00te3t702","country_code":"US","type":"education","lineage":["https://openalex.org/I165733156"]},{"id":"https://openalex.org/I137902535","display_name":"North Carolina State University","ror":"https://ror.org/04tj63d06","country_code":"US","type":"education","lineage":["https://openalex.org/I137902535"]}],"countries":["FR","US"],"is_corresponding":false,"raw_author_name":"Carla D. Savage","raw_affiliation_strings":["Department of Computer Science University of Georgia Athens, GA 30602, USA","Department of Computer Science North Carolina State University Raleigh, NC 27695-8206, USA","Laboratoire de Recherche en Informatique B\u00e2t 490, Universit\u00e9 Paris-Sud 91405 Orsay, FRANCE"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science University of Georgia Athens, GA 30602, USA","institution_ids":["https://openalex.org/I165733156"]},{"raw_affiliation_string":"Department of Computer Science North Carolina State University Raleigh, NC 27695-8206, USA","institution_ids":["https://openalex.org/I137902535"]},{"raw_affiliation_string":"Laboratoire de Recherche en Informatique B\u00e2t 490, Universit\u00e9 Paris-Sud 91405 Orsay, FRANCE","institution_ids":["https://openalex.org/I4210144804","https://openalex.org/I102197404"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5071772779"],"corresponding_institution_ids":["https://openalex.org/I102197404","https://openalex.org/I137902535","https://openalex.org/I165733156","https://openalex.org/I4210144804"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":8,"citation_normalized_percentile":{"value":0.42766419,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"5","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12508","display_name":"Functional Equations Stability Results","score":0.9861999750137329,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T12508","display_name":"Functional Equations Stability Results","score":0.9861999750137329,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/mathematics","display_name":"Mathematics","score":0.8686901330947876},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8482201099395752},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.708038330078125},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5867913365364075},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4615808129310608},{"id":"https://openalex.org/keywords/infinity","display_name":"Infinity","score":0.45115339756011963},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.4391050934791565},{"id":"https://openalex.org/keywords/square","display_name":"Square (algebra)","score":0.4206404983997345},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1190686821937561},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.11230283975601196}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8686901330947876},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8482201099395752},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.708038330078125},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5867913365364075},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4615808129310608},{"id":"https://openalex.org/C7321624","wikidata":"https://www.wikidata.org/wiki/Q205","display_name":"Infinity","level":2,"score":0.45115339756011963},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.4391050934791565},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.4206404983997345},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1190686821937561},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11230283975601196},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/1370","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1370","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v5i1r32/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"}],"best_oa_location":{"id":"doi:10.37236/1370","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1370","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v5i1r32/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":[{"id":"https://openalex.org/G1630494089","display_name":"Gray Codes, Efficient Generation, and Structure in          Combinatorial Families","funder_award_id":"9302505","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G625517339","display_name":"Structure, Generating, and Counting Problems in             Combinatorial Families","funder_award_id":"9622772","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6507447573","display_name":null,"funder_award_id":"several","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G848032724","display_name":null,"funder_award_id":"Science","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"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4253398483.pdf","grobid_xml":"https://content.openalex.org/works/W4253398483.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W1198302060","https://openalex.org/W1969474871","https://openalex.org/W1999942773","https://openalex.org/W2034564970","https://openalex.org/W2082637057","https://openalex.org/W2313024697","https://openalex.org/W2479102739","https://openalex.org/W2993539469","https://openalex.org/W3016843226","https://openalex.org/W4212799367","https://openalex.org/W4237125196","https://openalex.org/W4239614637","https://openalex.org/W4250083519","https://openalex.org/W6631343838","https://openalex.org/W6651579426"],"related_works":["https://openalex.org/W2306553012","https://openalex.org/W4251417359","https://openalex.org/W4229993731","https://openalex.org/W2381059884","https://openalex.org/W2372756075","https://openalex.org/W1979193983","https://openalex.org/W2220405813","https://openalex.org/W3013762056","https://openalex.org/W4287823680","https://openalex.org/W1983440566"],"abstract_inverted_index":{"Let":[0],"${\\bf":[1,12,20,34,70,103,151,249],"F}(n)$":[2,21,35,104,250],"be":[3,37],"a":[4,81,173,253,256],"family":[5,147],"of":[6,8,17,25,33,74,89,111,122,141,148,180,182,194,255,281],"partitions":[7,18,101,181],"$n$":[9,183,203],"and":[10,95,97,117,220,231,284],"let":[11],"F}(n,d)$":[13],"denote":[14],"the":[15,30,38,75,86,108,112,118,123,142,146,153,171,178,191,207,278,292,298,306],"set":[16],"in":[19,56,102,139,170,223],"with":[22,184,305],"Durfee":[23,31,76,113,124,154],"square":[24],"size":[26,110,121],"$d$.":[27],"We":[28,199,271],"define":[29],"polynomial":[32,39,77,155],"to":[36,205,287],"$P_{{\\bf":[40],"F},n}=":[41],"\\sum":[42],"|{\\bf":[43],"F}(n,d)|y^d$,":[44],"where":[45],"$":[46],"0":[47],"\\leq":[48,50,212,214],"d":[49,213],"\\lfloor":[51],"\\sqrt{n}":[52],"\\rfloor.$":[53],"The":[54],"work":[55],"this":[57,134,273],"paper":[58],"is":[59,93,216],"motivated":[60],"by":[61,129,237],"empirical":[62],"evidence":[63,242],"which":[64,189,234,251,302],"suggests":[65,244],"that":[66,85,144,201,245],"for":[67,105,145,166,196,228,246,265,291,297],"several":[68,247],"families":[69,248],"F}$,":[71],"all":[72,100,157],"roots":[73,158],"are":[78,226],"real.":[79,159],"Such":[80],"result":[82],"would":[83],"imply":[84],"corresponding":[87],"sequence":[88,208],"coefficients":[90],"$\\{|{\\bf":[91,209],"F}(n,d)|\\}$":[92],"log-concave":[94],"unimodal":[96],"that,":[98],"over":[99],"fixed":[106],"$n$,":[107],"average":[109],"square,":[114,125],"$a_{{\\bf":[115,229],"F}}(n)$,":[116,127],"most":[119,186,239],"likely":[120],"$m_{{\\bf":[126,232,259],"differ":[128,235],"less":[130],"than":[131],"1.":[132,240],"In":[133],"paper,":[135],"we":[136,161],"prove":[137,272],"results":[138],"support":[140],"conjecture":[143],"ordinary":[149],"partitions,":[150],"P}(n)$,":[152],"has":[156],"Specifically,":[160],"find":[162],"an":[163,266,275],"asymptotic":[164,279],"formula":[165],"$|{\\bf":[167,197,282,293],"P}(n,d)|$,":[168],"deriving":[169],"process":[172],"simple":[174],"upper":[175,192],"bound":[176,193],"on":[177],"number":[179],"at":[185,238],"$k$":[187],"parts":[188],"generalizes":[190],"Erd\u00f6s":[195],"P}(n)|$.":[198],"show":[200,285],"as":[202,262],"tends":[204],"infinity,":[206],"P}(n,d)|\\},\\":[210],"1":[211],"\\sqrt{n},$":[215],"asymptotically":[217,236],"normal,":[218],"unimodal,":[219],"log":[221],"concave;":[222],"addition,":[224],"formulas":[225],"found":[227],"P}}(n)$":[230,233],"Experimental":[241],"also":[243],"satisfy":[252],"recurrence":[254],"certain":[257],"form,":[258],"F}}(n)$":[260],"grows":[261],"$c":[263],"\\sqrt{n}$,":[264],"appropriate":[267],"constant":[268],"$c=c_{{\\bf":[269],"F}}$.":[270],"under":[274],"assumption":[276],"about":[277],"form":[280],"F}(n,d)|$":[283],"how":[286],"produce,":[288],"from":[289],"recurrences":[290],"F}(n,d)|$,":[294],"analytical":[295],"expressions":[296],"constants":[299],"$c_{{\\bf":[300],"F}}$":[301],"agree":[303],"numerically":[304],"observed":[307],"values.":[308]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-03-15T09:29:46.208133","created_date":"2022-05-12T00:00:00"}