{"id":"https://openalex.org/W1712117507","doi":"https://doi.org/10.37236/3053","title":"Nested Recursions, Simultaneous Parameters and Tree Superpositions","display_name":"Nested Recursions, Simultaneous Parameters and Tree Superpositions","publication_year":2014,"publication_date":"2014-03-10","ids":{"openalex":"https://openalex.org/W1712117507","doi":"https://doi.org/10.37236/3053","mag":"1712117507"},"language":"en","primary_location":{"id":"doi:10.37236/3053","is_oa":true,"landing_page_url":"https://doi.org/10.37236/3053","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v21i1p49/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/v21i1p49/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5030862985","display_name":"Abraham Isgur","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Abraham Isgur","raw_affiliation_strings":["Department of Mathematics\r\nUniversity of Toronto","Department of Mathematics#R##N#University of Toronto"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics\r\nUniversity of Toronto","institution_ids":[]},{"raw_affiliation_string":"Department of Mathematics#R##N#University of Toronto","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5047544753","display_name":"Vitaly Kuznetsov","orcid":null},"institutions":[{"id":"https://openalex.org/I36672615","display_name":"Courant Institute of Mathematical Sciences","ror":"https://ror.org/037tm7f56","country_code":"US","type":"education","lineage":["https://openalex.org/I36672615","https://openalex.org/I57206974"]},{"id":"https://openalex.org/I57206974","display_name":"New York University","ror":"https://ror.org/0190ak572","country_code":"US","type":"education","lineage":["https://openalex.org/I57206974"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Vitaly Kuznetsov","raw_affiliation_strings":["Courant Institute of Mathematical Sciences\r\nNew York University","Courant Institute of Mathematical Sciences#R##N#New York University"],"affiliations":[{"raw_affiliation_string":"Courant Institute of Mathematical Sciences\r\nNew York University","institution_ids":["https://openalex.org/I36672615","https://openalex.org/I57206974"]},{"raw_affiliation_string":"Courant Institute of Mathematical Sciences#R##N#New York University","institution_ids":["https://openalex.org/I36672615","https://openalex.org/I57206974"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5046343486","display_name":"Mustazee Rahman","orcid":"https://orcid.org/0000-0003-3071-9393"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Mustazee Rahman","raw_affiliation_strings":["Department of Mathematics\r\nUniversity of Toronto","Department of Mathematics#R##N#University of Toronto"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics\r\nUniversity of Toronto","institution_ids":[]},{"raw_affiliation_string":"Department of Mathematics#R##N#University of Toronto","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5086012769","display_name":"Stephen M. Tanny","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Stephen Tanny","raw_affiliation_strings":["Department of Mathematics\r\nUniversity of Toronto","Department of Mathematics#R##N#University of Toronto"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics\r\nUniversity of Toronto","institution_ids":[]},{"raw_affiliation_string":"Department of Mathematics#R##N#University of Toronto","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5030862985"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.04742453,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"21","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11269","display_name":"Algorithms and Data Compression","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11269","display_name":"Algorithms and Data Compression","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9975000023841858,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/arity","display_name":"Arity","score":0.9781187176704407},{"id":"https://openalex.org/keywords/recursion","display_name":"Recursion (computer science)","score":0.8827477097511292},{"id":"https://openalex.org/keywords/tree","display_name":"Tree (set theory)","score":0.7075175642967224},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.6228422522544861},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6045733690261841},{"id":"https://openalex.org/keywords/natural-number","display_name":"Natural number","score":0.5716738700866699},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.564440131187439},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.44217705726623535},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4300321042537689},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.19897034764289856}],"concepts":[{"id":"https://openalex.org/C110546421","wikidata":"https://www.wikidata.org/wiki/Q1315869","display_name":"Arity","level":2,"score":0.9781187176704407},{"id":"https://openalex.org/C168773036","wikidata":"https://www.wikidata.org/wiki/Q264164","display_name":"Recursion (computer science)","level":2,"score":0.8827477097511292},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.7075175642967224},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.6228422522544861},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6045733690261841},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.5716738700866699},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.564440131187439},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.44217705726623535},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4300321042537689},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.19897034764289856},{"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},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.37236/3053","is_oa":true,"landing_page_url":"https://doi.org/10.37236/3053","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v21i1p49/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:1301.5055","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1301.5055","pdf_url":"https://arxiv.org/pdf/1301.5055","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:1712117507","is_oa":true,"landing_page_url":"http://export.arxiv.org/pdf/1301.5055","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:CiteSeerX.psu:10.1.1.769.297","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.769.297","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/1301.5055.pdf","raw_type":"text"},{"id":"pmh:oai:durham-repository.worktribe.com:1290515","is_oa":true,"landing_page_url":"https://durham-repository.worktribe.com/output/1290515","pdf_url":null,"source":{"id":"https://openalex.org/S4306400188","display_name":"Durham Research Online (Durham University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I190082696","host_organization_name":"Durham University","host_organization_lineage":["https://openalex.org/I190082696"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Journal Article"},{"id":"doi:10.48550/arxiv.1301.5055","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1301.5055","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-journal"}],"best_oa_location":{"id":"doi:10.37236/3053","is_oa":true,"landing_page_url":"https://doi.org/10.37236/3053","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v21i1p49/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":[{"id":"https://openalex.org/G1845600299","display_name":null,"funder_award_id":"NSERC CGS","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"},{"id":"https://openalex.org/G3771973391","display_name":null,"funder_award_id":"NSERC PGS","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"},{"id":"https://openalex.org/G4918518531","display_name":null,"funder_award_id":"Rahman","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"},{"id":"https://openalex.org/G5463241825","display_name":null,"funder_award_id":"CGS grant","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"}],"funders":[{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1712117507.pdf","grobid_xml":"https://content.openalex.org/works/W1712117507.grobid-xml"},"referenced_works_count":9,"referenced_works":["https://openalex.org/W1554222672","https://openalex.org/W1848286548","https://openalex.org/W1884798717","https://openalex.org/W2030947627","https://openalex.org/W2037416325","https://openalex.org/W2131975287","https://openalex.org/W2763810095","https://openalex.org/W2963902573","https://openalex.org/W3101672871"],"related_works":["https://openalex.org/W2037416325","https://openalex.org/W2528938086","https://openalex.org/W2951604399","https://openalex.org/W2951661484","https://openalex.org/W2165946718","https://openalex.org/W2139153169","https://openalex.org/W2964232329","https://openalex.org/W1652586377","https://openalex.org/W2949076455","https://openalex.org/W3106318087","https://openalex.org/W997169918","https://openalex.org/W2257353082","https://openalex.org/W2008067984","https://openalex.org/W613729748","https://openalex.org/W2783809158","https://openalex.org/W2127926649","https://openalex.org/W180633690","https://openalex.org/W1571153713","https://openalex.org/W2509151057","https://openalex.org/W3094063364"],"abstract_inverted_index":{"We":[0,86,187],"apply":[1],"a":[2,67,78],"tree-based":[3,54],"methodology":[4],"to":[5,37,58,74,82,154,163,183],"solve":[6,166],"new,":[7],"very":[8],"broadly":[9],"defined":[10],"families":[11,90,124,139,170],"of":[12,15,52,110,125,136,144,156],"nested":[13,137,150],"recursions":[14,61,76],"the":[16,53,72,75,83,101,111,134,157],"general":[17,175,184],"form":[18],"$R(n)=\\sum_{t=1}^k":[19],"R(n-a_t-\\sum_{i=1}^{p}R(n-b_{ti}))$,":[20],"where":[21],"$a_t$":[22],"are":[23,26,31],"integers,":[24],"$b_{ti}$":[25],"natural":[27,32,68],"numbers,":[28],"and":[29,40,43,104,117,165,174],"$k,p$":[30],"numbers":[33],"that":[34,71,97,105],"we":[35,115,132,179],"use":[36],"denote":[38],"\"arity\"":[39],"\"order,\"":[41],"respectively,":[42],"with":[44,62,189],"some":[45],"specified":[46],"initial":[47],"conditions.":[48],"The":[49],"key":[50],"idea":[51],"solution":[55,73,135,145],"method":[56],"is":[57],"associate":[59],"such":[60],"infinite":[63],"labelled":[64],"trees":[65],"in":[66],"way":[69],"so":[70],"solves":[77],"counting":[79],"question":[80],"relating":[81],"corresponding":[84,112],"trees.":[85],"characterize":[87],"certain":[88],"recursion":[89,102,138,169],"within":[91,100],"$R(n)$":[92],"by":[93,140],"introducing":[94],"\"simultaneous":[95],"parameters\"":[96],"appear":[98],"both":[99],"itself":[103],"also":[106],"specify":[107],"structural":[108],"properties":[109],"tree.":[113],"First,":[114],"extend":[116,180],"unify":[118],"recently":[119],"discovered":[120],"results":[121,182],"concerning":[122],"two":[123,167],"arity":[126,172,185],"$k=2$,":[127],"order":[128,176],"$p=1$":[129],"recursions.":[130],"Next,":[131],"investigate":[133],"taking":[141],"linear":[142],"combinations":[143],"sequence":[146],"frequencies":[147],"for":[148,171],"simpler":[149],"recursions,":[151],"which":[152],"correspond":[153],"superpositions":[155],"associated":[158],"trees;":[159],"this":[160],"leads":[161],"us":[162],"identify":[164],"new":[168],"$k=2$":[173],"$p$.":[177],"Finally,":[178],"these":[181],"$k>2$.":[186],"conclude":[188],"several":[190],"related":[191],"open":[192],"problems.":[193]},"counts_by_year":[],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}