{"id":"https://openalex.org/W2125944388","doi":"https://doi.org/10.1017/s0963548310000192","title":"A Bijective Proof of a Theorem of Knuth","display_name":"A Bijective Proof of a Theorem of Knuth","publication_year":2010,"publication_date":"2010-07-21","ids":{"openalex":"https://openalex.org/W2125944388","doi":"https://doi.org/10.1017/s0963548310000192","mag":"2125944388"},"language":"en","primary_location":{"id":"doi:10.1017/s0963548310000192","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548310000192","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5037305170","display_name":"Hoda Bidkhori","orcid":"https://orcid.org/0000-0003-3895-5104"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"HODA BIDKHORI","raw_affiliation_strings":["Massachusetts Institute of Technology, Massachusetts Avenue, Cambridge, MA 02139, USA (e-mail:","Massachusetts institute of technology, massachusetts avenue, cambridge, ma 02139, usa (e-mail: bidkhori@mit.edu, skishore@mit.edu)#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Massachusetts Institute of Technology, Massachusetts Avenue, Cambridge, MA 02139, USA (e-mail:","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Massachusetts institute of technology, massachusetts avenue, cambridge, ma 02139, usa (e-mail: bidkhori@mit.edu, skishore@mit.edu)#TAB#","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5103364674","display_name":"Shaunak Kishore","orcid":null},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"SHAUNAK KISHORE","raw_affiliation_strings":["Massachusetts Institute of Technology, Massachusetts Avenue, Cambridge, MA 02139, USA (e-mail:","Massachusetts institute of technology, massachusetts avenue, cambridge, ma 02139, usa (e-mail: bidkhori@mit.edu, skishore@mit.edu)#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Massachusetts Institute of Technology, Massachusetts Avenue, Cambridge, MA 02139, USA (e-mail:","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Massachusetts institute of technology, massachusetts avenue, cambridge, ma 02139, usa (e-mail: bidkhori@mit.edu, skishore@mit.edu)#TAB#","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I63966007"],"apc_list":null,"apc_paid":null,"fwci":0.9386,"has_fulltext":false,"cited_by_count":10,"citation_normalized_percentile":{"value":0.72192321,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"20","issue":"1","first_page":"11","last_page":"25"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9994000196456909,"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":0.9994000196456909,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9987000226974487,"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":0.9965999722480774,"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/bijection","display_name":"Bijection","score":0.9351012706756592},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8628798723220825},{"id":"https://openalex.org/keywords/de-bruijn-sequence","display_name":"De Bruijn sequence","score":0.8552423715591431},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.809516429901123},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5633599162101746},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.4773590564727783},{"id":"https://openalex.org/keywords/binary-tree","display_name":"Binary tree","score":0.4617716073989868},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.38092178106307983}],"concepts":[{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.9351012706756592},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8628798723220825},{"id":"https://openalex.org/C170320093","wikidata":"https://www.wikidata.org/wiki/Q1953457","display_name":"De Bruijn sequence","level":2,"score":0.8552423715591431},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.809516429901123},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5633599162101746},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.4773590564727783},{"id":"https://openalex.org/C197855036","wikidata":"https://www.wikidata.org/wiki/Q380172","display_name":"Binary tree","level":2,"score":0.4617716073989868},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.38092178106307983}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1017/s0963548310000192","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548310000192","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":12,"referenced_works":["https://openalex.org/W1551177513","https://openalex.org/W1579213680","https://openalex.org/W1910083634","https://openalex.org/W1998018314","https://openalex.org/W2012276589","https://openalex.org/W2061798885","https://openalex.org/W2105321516","https://openalex.org/W2165513768","https://openalex.org/W2487501103","https://openalex.org/W2914659449","https://openalex.org/W2952803432","https://openalex.org/W4255987376"],"related_works":["https://openalex.org/W3198016531","https://openalex.org/W127787461","https://openalex.org/W1998517951","https://openalex.org/W2039773446","https://openalex.org/W2126473010","https://openalex.org/W2397939707","https://openalex.org/W2952259524","https://openalex.org/W2279732055","https://openalex.org/W2116880213","https://openalex.org/W2082836330"],"abstract_inverted_index":{"The":[0],"line":[1],"graph":[2,7],"G":[3,8,16,27,48],"of":[4,15,23,44,47,66,72,84,90,102,114],"a":[5,10,39,54,63,70,77,112],"directed":[6],"has":[9],"vertex":[11],"for":[12,20,41,53],"every":[13,21],"edge":[14,19],"and":[17,50,87,107],"an":[18],"path":[22],"length":[24,91],"2":[25,92],"in":[26],".":[28,95],"In":[29,58],"1967,":[30],"Knuth":[31],"used":[32],"the":[33,42,99,104],"Matrix":[34],"Tree":[35],"Theorem":[36],"to":[37],"prove":[38],"formula":[40],"number":[43],"spanning":[45],"trees":[46],",":[49],"he":[51],"asked":[52],"bijective":[55,64],"proof":[56,65],"[6].":[57],"this":[59,73],"paper,":[60],"we":[61,75,97],"give":[62],"Knuth's":[67],"formula.":[68],"As":[69],"result":[71,113],"proof,":[74],"find":[76],"bijection":[78],"between":[79],"binary":[80,88],"de":[81,108],"Bruijn":[82,109],"sequences":[83,89],"degree":[85],"n":[86,93],"\u22121":[94],"Finally,":[96],"determine":[98],"critical":[100],"groups":[101],"all":[103],"Kautz":[105],"graphs":[106],"graphs,":[110],"generalizing":[111],"Levine":[115],"[7].":[116]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
