{"id":"https://openalex.org/W1856850261","doi":"https://doi.org/10.37236/441","title":"$\\lambda$-Factorials of $n$","display_name":"$\\lambda$-Factorials of $n$","publication_year":2010,"publication_date":"2010-12-10","ids":{"openalex":"https://openalex.org/W1856850261","doi":"https://doi.org/10.37236/441","mag":"1856850261"},"language":"en","primary_location":{"id":"doi:10.37236/441","is_oa":true,"landing_page_url":"https://doi.org/10.37236/441","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v17i1r169/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":"preprint","indexed_in":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v17i1r169/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5045830002","display_name":"Yidong Sun","orcid":"https://orcid.org/0000-0002-6763-9912"},"institutions":[{"id":"https://openalex.org/I43313876","display_name":"Dalian Maritime University","ror":"https://ror.org/002b7nr53","country_code":"CN","type":"education","lineage":["https://openalex.org/I43313876"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yidong Sun","raw_affiliation_strings":["Department of Mathematics Dalian Maritime University, 116026 Dalian, P.R. China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics Dalian Maritime University, 116026 Dalian, P.R. China","institution_ids":["https://openalex.org/I43313876"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5102883163","display_name":"Jujuan Zhuang","orcid":"https://orcid.org/0000-0003-1483-4738"},"institutions":[{"id":"https://openalex.org/I43313876","display_name":"Dalian Maritime University","ror":"https://ror.org/002b7nr53","country_code":"CN","type":"education","lineage":["https://openalex.org/I43313876"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jujuan Zhuang","raw_affiliation_strings":["Department of Mathematics Dalian Maritime University, 116026 Dalian, P.R. China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics Dalian Maritime University, 116026 Dalian, P.R. China","institution_ids":["https://openalex.org/I43313876"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5045830002"],"corresponding_institution_ids":["https://openalex.org/I43313876"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.06197183,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":1.0,"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":1.0,"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/T11428","display_name":"Advanced Mathematical Identities","score":0.9975000023841858,"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/T10287","display_name":"Algebraic structures and combinatorial models","score":0.9840999841690063,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/combinatorics","display_name":"Combinatorics","score":0.7526689767837524},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6882517337799072},{"id":"https://openalex.org/keywords/lambda","display_name":"Lambda","score":0.6338014602661133},{"id":"https://openalex.org/keywords/binomial-coefficient","display_name":"Binomial coefficient","score":0.6332226991653442},{"id":"https://openalex.org/keywords/identity","display_name":"Identity (music)","score":0.5980123281478882},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.5877295732498169},{"id":"https://openalex.org/keywords/tree","display_name":"Tree (set theory)","score":0.5235937833786011},{"id":"https://openalex.org/keywords/fibonacci-number","display_name":"Fibonacci number","score":0.5011570453643799},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.4802406430244446},{"id":"https://openalex.org/keywords/generating-function","display_name":"Generating function","score":0.4369432330131531},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.4246053695678711},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4168154299259186},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.16697880625724792},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.05258181691169739}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7526689767837524},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6882517337799072},{"id":"https://openalex.org/C2778113609","wikidata":"https://www.wikidata.org/wiki/Q10897","display_name":"Lambda","level":2,"score":0.6338014602661133},{"id":"https://openalex.org/C34718186","wikidata":"https://www.wikidata.org/wiki/Q209875","display_name":"Binomial coefficient","level":2,"score":0.6332226991653442},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.5980123281478882},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.5877295732498169},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.5235937833786011},{"id":"https://openalex.org/C173734053","wikidata":"https://www.wikidata.org/wiki/Q47577","display_name":"Fibonacci number","level":2,"score":0.5011570453643799},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.4802406430244446},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.4369432330131531},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.4246053695678711},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4168154299259186},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.16697880625724792},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.05258181691169739},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.37236/441","is_oa":true,"landing_page_url":"https://doi.org/10.37236/441","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v17i1r169/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:arXiv.org:1007.1339","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1007.1339","pdf_url":"https://arxiv.org/pdf/1007.1339","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:1856850261","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1007.1339","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}],"best_oa_location":{"id":"doi:10.37236/441","is_oa":true,"landing_page_url":"https://doi.org/10.37236/441","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v17i1r169/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/G1141613155","display_name":null,"funder_award_id":"10801020","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2087396116","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2376276132","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320335787","funder_display_name":"Fundamental Research Funds for the Central Universities"},{"id":"https://openalex.org/G243061624","display_name":null,"funder_award_id":"2009QN070","funder_id":"https://openalex.org/F4320335787","funder_display_name":"Fundamental Research Funds for the Central Universities"},{"id":"https://openalex.org/G3085993365","display_name":null,"funder_award_id":"(Grant No.","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3317480652","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G37568934","display_name":null,"funder_award_id":"Grant","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5167091242","display_name":null,"funder_award_id":"No. 1","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5848258319","display_name":null,"funder_award_id":"0 and","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7726157001","display_name":null,"funder_award_id":"Grant No.","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G848032724","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8951484681","display_name":null,"funder_award_id":"Grant","funder_id":"https://openalex.org/F4320335787","funder_display_name":"Fundamental Research Funds for the Central Universities"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320335787","display_name":"Fundamental Research Funds for the Central Universities","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1856850261.pdf","grobid_xml":"https://content.openalex.org/works/W1856850261.grobid-xml"},"referenced_works_count":8,"referenced_works":["https://openalex.org/W1992551970","https://openalex.org/W2018392555","https://openalex.org/W2048936241","https://openalex.org/W2108017049","https://openalex.org/W2126209209","https://openalex.org/W2139825306","https://openalex.org/W2616697817","https://openalex.org/W2914659449"],"related_works":["https://openalex.org/W2064371883","https://openalex.org/W3103924409","https://openalex.org/W2463977018","https://openalex.org/W3131907284","https://openalex.org/W2975773228","https://openalex.org/W2952925895","https://openalex.org/W3044477640","https://openalex.org/W3049424050","https://openalex.org/W3106669014","https://openalex.org/W1584081507","https://openalex.org/W3181607128","https://openalex.org/W78898123","https://openalex.org/W2287774978","https://openalex.org/W2951775999","https://openalex.org/W3011158404","https://openalex.org/W2594833577","https://openalex.org/W3046095826","https://openalex.org/W2032433923","https://openalex.org/W2949961693","https://openalex.org/W2964082759"],"abstract_inverted_index":{"Recently,":[0],"by":[1,51,72],"the":[2,30,42,46,73,82,106],"Riordan":[3],"identity":[4,88],"related":[5],"to":[6,56],"tree":[7],"enumerations,":[8],"\\begin{align*}":[9,22],"\\sum_{k=0}^{n}\\binom{n}{k}(k+1)!(n+1)^{n-k}":[10],"=":[11,24],"(n+1)^{n+1},":[12],"\\end{align*}":[13,26],"Sun":[14],"and":[15,54,77,89,101,111],"Xu":[16],"have":[17],"derived":[18],"another":[19],"analogous":[20],"one,":[21],"\\sum_{k=0}^{n}\\binom{n}{k}D_{k+1}(n+1)^{n-k}":[23],"n^{n+1},":[25],"where":[27],"$D_{k}$":[28],"is":[29],"number":[31],"of":[32,48,61,85,99,109,117],"permutations":[33],"with":[34],"no":[35],"fixed":[36],"points":[37],"on":[38],"$\\{1,2,\\dots,":[39],"k\\}$.":[40],"In":[41],"paper,":[43],"we":[44,93],"utilize":[45],"$\\lambda$-factorials":[47,98],"$n$,":[49],"defined":[50],"Eriksen,":[52],"Freij":[53],"W$\\ddot{a}$stlund,":[55],"give":[57],"a":[58,69],"unified":[59],"generalization":[60],"these":[62],"two":[63,78],"identities.":[64],"We":[65],"provide":[66],"for":[67,97,114],"it":[68],"combinatorial":[70],"proof":[71],"functional":[74],"digraph":[75],"theory":[76],"algebraic":[79],"proofs.":[80],"Using":[81],"umbral":[83],"representation":[84],"our":[86],"generalized":[87],"Abel's":[90],"binomial":[91],"formula,":[92],"deduce":[94],"several":[95],"properties":[96],"$n$":[100],"establish":[102],"interesting":[103],"relations":[104],"between":[105],"generating":[107],"functions":[108],"general":[110],"exponential":[112],"types":[113],"any":[115],"sequence":[116],"numbers":[118],"or":[119],"polynomials.":[120]},"counts_by_year":[],"updated_date":"2026-04-18T07:56:08.524223","created_date":"2025-10-10T00:00:00"}