{"id":"https://openalex.org/W2166318576","doi":"https://doi.org/10.37236/1027","title":"Combinatorics of Partial Derivatives","display_name":"Combinatorics of Partial Derivatives","publication_year":2006,"publication_date":"2006-01-07","ids":{"openalex":"https://openalex.org/W2166318576","doi":"https://doi.org/10.37236/1027","mag":"2166318576"},"language":"en","primary_location":{"id":"doi:10.37236/1027","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1027","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v13i1r1/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/v13i1r1/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5111953241","display_name":"Michael Hardy","orcid":null},"institutions":[{"id":"https://openalex.org/I130238516","display_name":"University of Minnesota","ror":"https://ror.org/017zqws13","country_code":"US","type":"education","lineage":["https://openalex.org/I130238516"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Michael Hardy","raw_affiliation_strings":["School of Mathematics University of Minnesota, Minneapolis, MN 55455, USA"],"affiliations":[{"raw_affiliation_string":"School of Mathematics University of Minnesota, Minneapolis, MN 55455, USA","institution_ids":["https://openalex.org/I130238516"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5111953241"],"corresponding_institution_ids":["https://openalex.org/I130238516"],"apc_list":null,"apc_paid":null,"fwci":4.2572,"has_fulltext":true,"cited_by_count":150,"citation_normalized_percentile":{"value":0.95115789,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.996999979019165,"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":0.996999979019165,"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/T12516","display_name":"Advanced Mathematical Theories and Applications","score":0.9771999716758728,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":0.9760000109672546,"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/mathematics","display_name":"Mathematics","score":0.8358296155929565},{"id":"https://openalex.org/keywords/partial-derivative","display_name":"Partial derivative","score":0.7016573548316956},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6674879193305969},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.6251227855682373},{"id":"https://openalex.org/keywords/enumeration","display_name":"Enumeration","score":0.5675520896911621},{"id":"https://openalex.org/keywords/cumulant","display_name":"Cumulant","score":0.5521194934844971},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.4918856620788574},{"id":"https://openalex.org/keywords/derivative","display_name":"Derivative (finance)","score":0.4907582700252533},{"id":"https://openalex.org/keywords/differential-operator","display_name":"Differential operator","score":0.46436071395874023},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.4602578580379486},{"id":"https://openalex.org/keywords/partially-ordered-set","display_name":"Partially ordered set","score":0.42929619550704956},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.315094530582428},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.11502107977867126},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.06409162282943726},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.0591374933719635}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8358296155929565},{"id":"https://openalex.org/C53846429","wikidata":"https://www.wikidata.org/wiki/Q186475","display_name":"Partial derivative","level":2,"score":0.7016573548316956},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6674879193305969},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.6251227855682373},{"id":"https://openalex.org/C156340839","wikidata":"https://www.wikidata.org/wiki/Q2704791","display_name":"Enumeration","level":2,"score":0.5675520896911621},{"id":"https://openalex.org/C172686274","wikidata":"https://www.wikidata.org/wiki/Q746007","display_name":"Cumulant","level":2,"score":0.5521194934844971},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.4918856620788574},{"id":"https://openalex.org/C111771559","wikidata":"https://www.wikidata.org/wiki/Q66295","display_name":"Derivative (finance)","level":2,"score":0.4907582700252533},{"id":"https://openalex.org/C70915906","wikidata":"https://www.wikidata.org/wiki/Q1058681","display_name":"Differential operator","level":2,"score":0.46436071395874023},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.4602578580379486},{"id":"https://openalex.org/C180645754","wikidata":"https://www.wikidata.org/wiki/Q474715","display_name":"Partially ordered set","level":2,"score":0.42929619550704956},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.315094530582428},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11502107977867126},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.06409162282943726},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0591374933719635},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C106159729","wikidata":"https://www.wikidata.org/wiki/Q2294553","display_name":"Financial economics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.37236/1027","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1027","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v13i1r1/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:10.1.1.238.1077","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.238.1077","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/math/0601149v1.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.87.270","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.87.270","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.emis.de/journals/EJC/Volume_13/PDF/v13i1r1.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/1027","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1027","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v13i1r1/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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2166318576.pdf","grobid_xml":"https://content.openalex.org/works/W2166318576.grobid-xml"},"referenced_works_count":18,"referenced_works":["https://openalex.org/W157784468","https://openalex.org/W186230741","https://openalex.org/W326772095","https://openalex.org/W1489988510","https://openalex.org/W1527870453","https://openalex.org/W1551177513","https://openalex.org/W1862031973","https://openalex.org/W1986022039","https://openalex.org/W2004162357","https://openalex.org/W2004170125","https://openalex.org/W2077510990","https://openalex.org/W2078762661","https://openalex.org/W2144232853","https://openalex.org/W2161099683","https://openalex.org/W4249797114","https://openalex.org/W6600127060","https://openalex.org/W6755020716","https://openalex.org/W6847347038"],"related_works":["https://openalex.org/W108845024","https://openalex.org/W4300125793","https://openalex.org/W2578985712","https://openalex.org/W2406961474","https://openalex.org/W2143566611","https://openalex.org/W2952599326","https://openalex.org/W1561106399","https://openalex.org/W2028787035","https://openalex.org/W4295203081","https://openalex.org/W2000758421"],"abstract_inverted_index":{"The":[0,93],"natural":[1],"forms":[2,65],"of":[3,11,15,24,59,66,78,95,101,126],"the":[4,8,21,28,44,96,122],"Leibniz":[5],"rule":[6],"for":[7,20],"$k$th":[9,22],"derivative":[10,23],"a":[12,25,107],"product":[13],"and":[14],"Fa\u00e0":[16,102],"di":[17,103],"Bruno's":[18,104],"formula":[19,105],"composition":[26],"involve":[27],"differential":[29],"operator":[30],"$\\partial^k/\\partial":[31],"x_1":[32],"\\cdots":[33],"\\partial":[34],"x_k$":[35],"rather":[36],"than":[37],"$d^k/dx^k$,":[38],"with":[39],"no":[40],"assumptions":[41],"about":[42],"whether":[43],"variables":[45,72,85],"$x_1,\\dots,x_k$":[46],"are":[47,73,75,86,91],"all":[48,51,84,89],"distinct,":[49],"or":[50,53],"identical,":[52],"partitioned":[54],"into":[55],"several":[56],"distinguishable":[57],"classes":[58],"indistinguishable":[60,74,79],"variables.":[61],"Coefficients":[62],"appearing":[63],"in":[64,69,98],"these":[67],"identities":[68],"which":[70],"some":[71],"just":[76],"multiplicities":[77,97],"terms":[80],"(in":[81],"particular,":[82],"if":[83],"distinct":[87],"then":[88],"coefficients":[90],"1).":[92],"computation":[94],"this":[99],"generalization":[100],"is":[106],"combinatorial":[108],"enumeration":[109],"problem":[110],"that,":[111],"although":[112],"completely":[113],"elementary,":[114],"seems":[115],"to":[116,124],"have":[117],"been":[118],"neglected.":[119],"We":[120],"apply":[121],"results":[123],"cumulants":[125],"probability":[127],"distributions.":[128]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":8},{"year":2024,"cited_by_count":8},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":9},{"year":2021,"cited_by_count":20},{"year":2020,"cited_by_count":7},{"year":2019,"cited_by_count":12},{"year":2018,"cited_by_count":9},{"year":2017,"cited_by_count":7},{"year":2016,"cited_by_count":6},{"year":2015,"cited_by_count":12},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":12},{"year":2012,"cited_by_count":7}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}