{"id":"https://openalex.org/W2887452462","doi":"https://doi.org/10.37236/8585","title":"Families of Major Index Distributions: Closed Forms and Unimodality","display_name":"Families of Major Index Distributions: Closed Forms and Unimodality","publication_year":2019,"publication_date":"2019-09-27","ids":{"openalex":"https://openalex.org/W2887452462","doi":"https://doi.org/10.37236/8585","mag":"2887452462"},"language":"en","primary_location":{"id":"doi:10.37236/8585","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8585","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v26i3p58/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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v26i3p58/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5068769720","display_name":"William J. Keith","orcid":"https://orcid.org/0000-0002-5109-0216"},"institutions":[{"id":"https://openalex.org/I11957088","display_name":"Michigan Technological University","ror":"https://ror.org/0036rpn28","country_code":"US","type":"education","lineage":["https://openalex.org/I11957088"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"William J. Keith","raw_affiliation_strings":["Michigan Technological University"],"affiliations":[{"raw_affiliation_string":"Michigan Technological University","institution_ids":["https://openalex.org/I11957088"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5068769720"],"corresponding_institution_ids":["https://openalex.org/I11957088"],"apc_list":null,"apc_paid":null,"fwci":0.49953052,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.54740667,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":93},"biblio":{"volume":"26","issue":"3","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/T11680","display_name":"Advanced Algebra and Geometry","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11716","display_name":"Random Matrices and Applications","score":0.998199999332428,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/unimodality","display_name":"Unimodality","score":0.9567639827728271},{"id":"https://openalex.org/keywords/young-tableau","display_name":"Young tableau","score":0.807928740978241},{"id":"https://openalex.org/keywords/realization","display_name":"Realization (probability)","score":0.6834135055541992},{"id":"https://openalex.org/keywords/lambda","display_name":"Lambda","score":0.6805670857429504},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6601916551589966},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6115224361419678},{"id":"https://openalex.org/keywords/distribution","display_name":"Distribution (mathematics)","score":0.6058926582336426},{"id":"https://openalex.org/keywords/index","display_name":"Index (typography)","score":0.534087598323822},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4854457974433899},{"id":"https://openalex.org/keywords/principal","display_name":"Principal (computer security)","score":0.43301868438720703},{"id":"https://openalex.org/keywords/symmetric-function","display_name":"Symmetric function","score":0.4268075227737427},{"id":"https://openalex.org/keywords/generating-function","display_name":"Generating function","score":0.4257115423679352},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4154420793056488},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.189884215593338},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1219111979007721},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.10432374477386475},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08471435308456421}],"concepts":[{"id":"https://openalex.org/C21596040","wikidata":"https://www.wikidata.org/wiki/Q2896771","display_name":"Unimodality","level":2,"score":0.9567639827728271},{"id":"https://openalex.org/C165368118","wikidata":"https://www.wikidata.org/wiki/Q2166280","display_name":"Young tableau","level":2,"score":0.807928740978241},{"id":"https://openalex.org/C2781089630","wikidata":"https://www.wikidata.org/wiki/Q21856745","display_name":"Realization (probability)","level":2,"score":0.6834135055541992},{"id":"https://openalex.org/C2778113609","wikidata":"https://www.wikidata.org/wiki/Q10897","display_name":"Lambda","level":2,"score":0.6805670857429504},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6601916551589966},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6115224361419678},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.6058926582336426},{"id":"https://openalex.org/C2777382242","wikidata":"https://www.wikidata.org/wiki/Q6017816","display_name":"Index (typography)","level":2,"score":0.534087598323822},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4854457974433899},{"id":"https://openalex.org/C144559511","wikidata":"https://www.wikidata.org/wiki/Q2986279","display_name":"Principal (computer security)","level":2,"score":0.43301868438720703},{"id":"https://openalex.org/C3746008","wikidata":"https://www.wikidata.org/wiki/Q981351","display_name":"Symmetric function","level":2,"score":0.4268075227737427},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.4257115423679352},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4154420793056488},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.189884215593338},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1219111979007721},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.10432374477386475},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08471435308456421},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","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/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C136764020","wikidata":"https://www.wikidata.org/wiki/Q466","display_name":"World Wide Web","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.37236/8585","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8585","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v26i3p58/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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1808.01362","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1808.01362","pdf_url":"https://arxiv.org/pdf/1808.01362","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},{"id":"mag:2887452462","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1808.01362.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:digitalcommons.mtu.edu:michigantech-p-2223","is_oa":true,"landing_page_url":"https://digitalcommons.mtu.edu/michigantech-p/1225","pdf_url":null,"source":{"id":"https://openalex.org/S4377196391","display_name":"Digital Commons - Michigan Tech (Michigan Technological University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I11957088","host_organization_name":"Michigan Technological University","host_organization_lineage":["https://openalex.org/I11957088"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nd","license_id":"https://openalex.org/licenses/cc-by-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Michigan Tech Publications","raw_type":"text"},{"id":"doi:10.48550/arxiv.1808.01362","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1808.01362","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.37236/8585","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8585","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v26i3p58/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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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":[{"id":"https://openalex.org/F4320309636","display_name":"University of Minnesota","ror":"https://ror.org/03grvy078"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2887452462.pdf","grobid_xml":"https://content.openalex.org/works/W2887452462.grobid-xml"},"referenced_works_count":17,"referenced_works":["https://openalex.org/W1969719825","https://openalex.org/W1985237288","https://openalex.org/W2035734811","https://openalex.org/W2044185619","https://openalex.org/W2075282864","https://openalex.org/W2080623591","https://openalex.org/W2094042760","https://openalex.org/W2134984950","https://openalex.org/W2165384925","https://openalex.org/W2229099293","https://openalex.org/W2536100367","https://openalex.org/W2733286345","https://openalex.org/W2790716536","https://openalex.org/W2962709632","https://openalex.org/W3013625416","https://openalex.org/W3013628723","https://openalex.org/W6670909874"],"related_works":["https://openalex.org/W2999121026","https://openalex.org/W2776903931","https://openalex.org/W210894620","https://openalex.org/W2963051726","https://openalex.org/W2803390059","https://openalex.org/W2964223138","https://openalex.org/W2276893634","https://openalex.org/W2807268134","https://openalex.org/W1938752131","https://openalex.org/W2569834454","https://openalex.org/W3208589802","https://openalex.org/W3158871452","https://openalex.org/W3083596116","https://openalex.org/W3102290155","https://openalex.org/W2962683514","https://openalex.org/W3089784040","https://openalex.org/W2963789022","https://openalex.org/W2809468009","https://openalex.org/W2971524145","https://openalex.org/W3103097199"],"abstract_inverted_index":{"Closed":[0],"forms":[1],"for":[2,34],"$f_{\\lambda,i}":[3],"(q)":[4],":=":[5],"\\sum_{\\tau":[6],"\\in":[7],"SYT(\\lambda)":[8],":":[9],"des(\\tau)":[10],"=":[11],"i}":[12],"q^{maj(\\tau)}$,":[13],"the":[14,17,46,64,86,90,115,118],"distribution":[15,116],"of":[16,24,30,38,56,72,82,92,105,117,125],"major":[18,119],"index":[19,120],"over":[20,121],"standard":[21],"Young":[22],"tableaux":[23],"given":[25,123],"shapes":[26],"and":[27,40,58,74],"specified":[28],"number":[29,91],"descents,":[31],"are":[32,66],"established":[33,62],"a":[35,50,54,70,111,122],"large":[36],"collection":[37],"$\\lambda$":[39],"$i$.":[41],"Of":[42],"particular":[43],"interest":[44],"is":[45,94,98],"family":[47],"that":[48,100],"gives":[49],"positive":[51],"answer":[52],"to":[53],"question":[55],"Sagan":[57],"collaborators.":[59],"All":[60],"formulas":[61],"in":[63],"paper":[65],"unimodal,":[67],"most":[68],"by":[69],"result":[71],"Kirillov":[73],"Reshetikhin.":[75],"Many":[76],"can":[77],"be":[78],"identified":[79],"as":[80,114],"specializations":[81],"Schur":[83,107],"functions":[84],"via":[85],"Jacobi-Trudi":[87],"identities.":[88],"If":[89],"arguments":[93],"sufficiently":[95],"large,":[96],"it":[97],"shown":[99],"any":[101,106],"finite":[102],"principal":[103],"specialization":[104],"function":[108],"$s_\\lambda(1,q,q^2,\\dots,q^{n-1})$":[109],"has":[110],"combinatorial":[112],"realization":[113],"set":[124],"tableaux.":[126]},"counts_by_year":[{"year":2021,"cited_by_count":1}],"updated_date":"2026-02-09T09:26:11.010843","created_date":"2025-10-10T00:00:00"}