Method for Constructing Bijections for Classical Partition Identities

Method for Constructing Bijections for Classical Partition Identities

We sketch the construction of a bijection between the partitions of n with parts congruent to 1 or 4 (mod 5) and the partitions of n with parts differing by at least 2. This bijection is obtained by a cut-and-paste procedure that starts with a partition in one class and ends with a partition in the...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Proceedings of the National Academy of Sciences - PNAS 1981-04, Vol.78 (4), p.2026-2028
Hauptverfasser: Garsia, A. M., Milne, S. C.
Format: Artikel
Sprache:eng
Schlagworte:
Combinatorial permutations
Combinatorics
Mathematical sets
Odd numbers
Physical Sciences: Mathematics
Symbolism
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
container_end_page 2028
container_issue 4
container_start_page 2026
container_title Proceedings of the National Academy of Sciences - PNAS
container_volume 78
creator Garsia, A. M.
Milne, S. C.
description We sketch the construction of a bijection between the partitions of n with parts congruent to 1 or 4 (mod 5) and the partitions of n with parts differing by at least 2. This bijection is obtained by a cut-and-paste procedure that starts with a partition in one class and ends with a partition in the other class. The whole construction is a combination of a bijection discovered quite early by Schur and two bijections of our own. A basic principle concerning pairs of involutions provides the key for connecting all these bijections. It appears that our methods lead to an algorithm for constructing bijections for other identities of Rogers-Ramanujan type such as the Gordon identities.
doi_str_mv 10.1073/pnas.78.4.2026
format Article
fullrecord <record><control><sourceid>jstor_pubme</sourceid><recordid>TN_cdi_pubmedcentral_primary_oai_pubmedcentral_nih_gov_319275</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><jstor_id>9821</jstor_id><sourcerecordid>9821</sourcerecordid><originalsourceid>FETCH-LOGICAL-c389t-6e1ae920136985d17b65e4cb722fbfd8eeac2188b509acdecbe8d898897a9493</originalsourceid><addsrcrecordid>eNp9kM1v1DAQxS0EokvhisQBlBOcEsYficcHDrAqUKkIDr1bjjNps8rGi-0g-O9JtEspF04zmvd7M6PH2HMOFQct3x4mlyqNlaoEiOYB23AwvGyUgYdsAyB0iUqoM_YkpR0AmBrhMTvjTW0kgNqwiy-Ub0NX9CEW2zClHGefh-mm-DDsaOmW0VEbXUqDd2PxzcU8rEJx2dG0tpSeske9GxM9O9Vzdv3x4nr7ubz6-uly-_6q9BJNLhvijowALhuDdcd129SkfKuF6Nu-QyLnBUdsazDOd-Rbwg4NotHOKCPP2bvj2sPc7qnzy_noRnuIw97FXza4wf6rTMOtvQk_rORG6Hrxvzn5Y_g-U8p2PyRP4-gmCnOyWsoaFCIuZHUkfQwpRervjnCwa_B2Dd5qtMquwS-GV_df-4ufkl6AlydgNf6R7y94_T_d9vM4ZvqZF_DFEdylHOIdaVBw-RvmbqF1</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>733504888</pqid></control><display><type>article</type><title>Method for Constructing Bijections for Classical Partition Identities</title><source>Jstor Complete Legacy</source><source>PubMed Central</source><source>Alma/SFX Local Collection</source><source>Free Full-Text Journals in Chemistry</source><creator>Garsia, A. M. ; Milne, S. C.</creator><creatorcontrib>Garsia, A. M. ; Milne, S. C.</creatorcontrib><description>We sketch the construction of a bijection between the partitions of n with parts congruent to 1 or 4 (mod 5) and the partitions of n with parts differing by at least 2. This bijection is obtained by a cut-and-paste procedure that starts with a partition in one class and ends with a partition in the other class. The whole construction is a combination of a bijection discovered quite early by Schur and two bijections of our own. A basic principle concerning pairs of involutions provides the key for connecting all these bijections. It appears that our methods lead to an algorithm for constructing bijections for other identities of Rogers-Ramanujan type such as the Gordon identities.</description><identifier>ISSN: 0027-8424</identifier><identifier>EISSN: 1091-6490</identifier><identifier>DOI: 10.1073/pnas.78.4.2026</identifier><identifier>PMID: 16593004</identifier><language>eng</language><publisher>United States: National Academy of Sciences of the United States of America</publisher><subject>Combinatorial permutations ; Combinatorics ; Mathematical sets ; Odd numbers ; Physical Sciences: Mathematics ; Symbolism</subject><ispartof>Proceedings of the National Academy of Sciences - PNAS, 1981-04, Vol.78 (4), p.2026-2028</ispartof><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c389t-6e1ae920136985d17b65e4cb722fbfd8eeac2188b509acdecbe8d898897a9493</citedby></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Uhttp://www.pnas.org/content/78/4.cover.gif</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/9821$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/9821$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>230,314,723,776,780,799,881,27901,27902,53766,53768,57992,58225</link.rule.ids><backlink>$$Uhttps://www.ncbi.nlm.nih.gov/pubmed/16593004$$D View this record in MEDLINE/PubMed$$Hfree_for_read</backlink></links><search><creatorcontrib>Garsia, A. M.</creatorcontrib><creatorcontrib>Milne, S. C.</creatorcontrib><title>Method for Constructing Bijections for Classical Partition Identities</title><title>Proceedings of the National Academy of Sciences - PNAS</title><addtitle>Proc Natl Acad Sci U S A</addtitle><description>We sketch the construction of a bijection between the partitions of n with parts congruent to 1 or 4 (mod 5) and the partitions of n with parts differing by at least 2. This bijection is obtained by a cut-and-paste procedure that starts with a partition in one class and ends with a partition in the other class. The whole construction is a combination of a bijection discovered quite early by Schur and two bijections of our own. A basic principle concerning pairs of involutions provides the key for connecting all these bijections. It appears that our methods lead to an algorithm for constructing bijections for other identities of Rogers-Ramanujan type such as the Gordon identities.</description><subject>Combinatorial permutations</subject><subject>Combinatorics</subject><subject>Mathematical sets</subject><subject>Odd numbers</subject><subject>Physical Sciences: Mathematics</subject><subject>Symbolism</subject><issn>0027-8424</issn><issn>1091-6490</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1981</creationdate><recordtype>article</recordtype><recordid>eNp9kM1v1DAQxS0EokvhisQBlBOcEsYficcHDrAqUKkIDr1bjjNps8rGi-0g-O9JtEspF04zmvd7M6PH2HMOFQct3x4mlyqNlaoEiOYB23AwvGyUgYdsAyB0iUqoM_YkpR0AmBrhMTvjTW0kgNqwiy-Ub0NX9CEW2zClHGefh-mm-DDsaOmW0VEbXUqDd2PxzcU8rEJx2dG0tpSeske9GxM9O9Vzdv3x4nr7ubz6-uly-_6q9BJNLhvijowALhuDdcd129SkfKuF6Nu-QyLnBUdsazDOd-Rbwg4NotHOKCPP2bvj2sPc7qnzy_noRnuIw97FXza4wf6rTMOtvQk_rORG6Hrxvzn5Y_g-U8p2PyRP4-gmCnOyWsoaFCIuZHUkfQwpRervjnCwa_B2Dd5qtMquwS-GV_df-4ufkl6AlydgNf6R7y94_T_d9vM4ZvqZF_DFEdylHOIdaVBw-RvmbqF1</recordid><startdate>19810401</startdate><enddate>19810401</enddate><creator>Garsia, A. M.</creator><creator>Milne, S. C.</creator><general>National Academy of Sciences of the United States of America</general><general>National Acad Sciences</general><scope>NPM</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7X8</scope><scope>5PM</scope></search><sort><creationdate>19810401</creationdate><title>Method for Constructing Bijections for Classical Partition Identities</title><author>Garsia, A. M. ; Milne, S. C.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c389t-6e1ae920136985d17b65e4cb722fbfd8eeac2188b509acdecbe8d898897a9493</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1981</creationdate><topic>Combinatorial permutations</topic><topic>Combinatorics</topic><topic>Mathematical sets</topic><topic>Odd numbers</topic><topic>Physical Sciences: Mathematics</topic><topic>Symbolism</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Garsia, A. M.</creatorcontrib><creatorcontrib>Milne, S. C.</creatorcontrib><collection>PubMed</collection><collection>CrossRef</collection><collection>MEDLINE - Academic</collection><collection>PubMed Central (Full Participant titles)</collection><jtitle>Proceedings of the National Academy of Sciences - PNAS</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Garsia, A. M.</au><au>Milne, S. C.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Method for Constructing Bijections for Classical Partition Identities</atitle><jtitle>Proceedings of the National Academy of Sciences - PNAS</jtitle><addtitle>Proc Natl Acad Sci U S A</addtitle><date>1981-04-01</date><risdate>1981</risdate><volume>78</volume><issue>4</issue><spage>2026</spage><epage>2028</epage><pages>2026-2028</pages><issn>0027-8424</issn><eissn>1091-6490</eissn><abstract>We sketch the construction of a bijection between the partitions of n with parts congruent to 1 or 4 (mod 5) and the partitions of n with parts differing by at least 2. This bijection is obtained by a cut-and-paste procedure that starts with a partition in one class and ends with a partition in the other class. The whole construction is a combination of a bijection discovered quite early by Schur and two bijections of our own. A basic principle concerning pairs of involutions provides the key for connecting all these bijections. It appears that our methods lead to an algorithm for constructing bijections for other identities of Rogers-Ramanujan type such as the Gordon identities.</abstract><cop>United States</cop><pub>National Academy of Sciences of the United States of America</pub><pmid>16593004</pmid><doi>10.1073/pnas.78.4.2026</doi><tpages>3</tpages><oa>free_for_read</oa></addata></record>
fulltext fulltext
identifier ISSN: 0027-8424
ispartof Proceedings of the National Academy of Sciences - PNAS, 1981-04, Vol.78 (4), p.2026-2028
issn 0027-8424
1091-6490
language eng
recordid cdi_pubmedcentral_primary_oai_pubmedcentral_nih_gov_319275
source Jstor Complete Legacy; PubMed Central; Alma/SFX Local Collection; Free Full-Text Journals in Chemistry
subjects Combinatorial permutations
Combinatorics
Mathematical sets
Odd numbers
Physical Sciences: Mathematics
Symbolism
title Method for Constructing Bijections for Classical Partition Identities
url https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-02-09T13%3A57%3A31IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-jstor_pubme&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Method%20for%20Constructing%20Bijections%20for%20Classical%20Partition%20Identities&rft.jtitle=Proceedings%20of%20the%20National%20Academy%20of%20Sciences%20-%20PNAS&rft.au=Garsia,%20A.%20M.&rft.date=1981-04-01&rft.volume=78&rft.issue=4&rft.spage=2026&rft.epage=2028&rft.pages=2026-2028&rft.issn=0027-8424&rft.eissn=1091-6490&rft_id=info:doi/10.1073/pnas.78.4.2026&rft_dat=%3Cjstor_pubme%3E9821%3C/jstor_pubme%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=733504888&rft_id=info:pmid/16593004&rft_jstor_id=9821&rfr_iscdi=true