A Combinatorial Approach Towards Ramanujan's Partition Congruences Modulo 5, 7 and 11 In Terms Of Its Corresponding Cranks

A Combinatorial Approach Towards Ramanujan's Partition Congruences Modulo 5, 7 and 11 In Terms Of Its Corresponding Cranks | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A Combinatorial Approach Towards Ramanujan's Partition Congruences Modulo 5, 7 and 11 In Terms Of Its Corresponding Cranks Subham De This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3325415/v2 This work is licensed under a CC BY 4.0 License Status: Posted Version 2 posted You are reading this latest preprint version Show more versions Abstract The primary purpose of this article is to provide an explicit interpretation of \textit{Ramanujan}'s Partition Congruences, namely, $p(5n+4)\equiv 0\mbox{ }(mod\mbox{ }5)$, $p(7n+5)\equiv 0\mbox{ }(mod\mbox{ }7)$ and, $p(11n+6)\equiv 0\mbox{ }(mod\mbox{ }11)$, $p(n)$ being the \textit{partition function} corresponding to any positive integer $n$, in terms of their corresponding \textit{Cranks} using various combinatorial arguments implemented by \textit{Dyson}, \textit{Atkin}, \textit{Swinnerton-Dyer} and later by \textit{Andrews} and, \textit{Garvan} to justify all the necessary concepts pertaining to this topic. 2020 MSC: Primary 11-02, 11P81, 11P82, 11P83, 11P84. Secondary 11B75, 05A16, 05A17, 05A30. Partition Rank Crank Vector Partition q-Series Ramanujan’s Theta Function Jacobi-Triple Product Identity Euler’s Pentagonal Theorem Successive Rank Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 2 posted You are reading this latest preprint version Show more versions Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3325415","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":331392682,"identity":"cca7a2b8-9c23-454a-a013-dd610d9b4d5c","order_by":0,"name":"Subham De","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/0lEQVRIiWNgGAWjYFACHjYg8U+Ojb35gMEHIJONnTgtB4z5eI4lFM4AaWEmUkviPIkcg888IAFCWuRn5B578DPnjjGbRI7hZptf2+T5mBkYP3zMwa3F4EZeumHvtmdybDzPio1z+24btjEzMEvO3IZHi0SOmQTvNmZjNvbkbca5PbcZgVrYmHnxaJGfkWMm+Xcbc2IbQ4L5b8ue2/YEtTDcyDGT5t12OLGNI8XAmOHH7USCWgzOvEuTlt2WZswGDGTD3obbyW3MjM14/SLfnntM8u02Gzn5dmBU/vhz23Z+e/PBDx/xOQwFMLaByQZi1YPAH1IUj4JRMApGwUgBAIsDUVSuf78OAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0009-0001-3265-4354","institution":"Indian Institute of Technology Delhi","correspondingAuthor":true,"prefix":"","firstName":"Subham","middleName":"","lastName":"De","suffix":""}],"badges":[],"createdAt":"2023-09-04 18:29:18","currentVersionCode":2,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-3325415/v2","doiUrl":"https://doi.org/10.21203/rs.3.rs-3325415/v2","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":62129852,"identity":"e4cc9b16-cb7c-4364-9df4-10ed88f3e58d","added_by":"auto","created_at":"2024-08-09 15:28:19","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":343518,"visible":true,"origin":"","legend":"","description":"","filename":"Partitions.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3325415/v2_covered_f5ee0296-76b7-482b-98db-871aa90cf436.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"A Combinatorial Approach Towards Ramanujan's Partition Congruences Modulo 5, 7 and 11 In Terms Of Its Corresponding Cranks","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Partition, Rank, Crank, Vector Partition, q-Series, Ramanujan’s Theta Function, Jacobi-Triple Product Identity, Euler’s Pentagonal Theorem, Successive Rank","lastPublishedDoi":"10.21203/rs.3.rs-3325415/v2","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3325415/v2","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe primary purpose of this article is to provide an explicit interpretation of \\textit{Ramanujan}'s Partition Congruences, namely, $p(5n+4)\\equiv 0\\mbox{ }(mod\\mbox{ }5)$, $p(7n+5)\\equiv 0\\mbox{ }(mod\\mbox{ }7)$ and, $p(11n+6)\\equiv 0\\mbox{ }(mod\\mbox{ }11)$, $p(n)$ being the \\textit{partition function} corresponding to any positive integer $n$, in terms of their corresponding \\textit{Cranks} using various combinatorial arguments implemented by \\textit{Dyson}, \\textit{Atkin}, \\textit{Swinnerton-Dyer} and later by \\textit{Andrews} and, \\textit{Garvan} to justify all the necessary concepts pertaining to this topic.\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003e2020 MSC: Primary 11-02, 11P81, 11P82, 11P83, 11P84. 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