A combinatorial approach towards Ramanujan’s partition congruences modulo 5, 7 and 11 in terms of its corresponding cranks
Subham Desubham581994@gmail.comDepartment of MathematicsHauz KhasIndian Institute of Technology DelhiNew Delhi, Delhi, 110016, India0009-0001-3265-4354View full profile →
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- Received:
- 01 Feb 2025
- Published Online:
- 30 Mar 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JDMSC-2416
- Pages:
- 2685–2698
Abstract
This paper presents a combinatorial interpretation of Ramanujan’s celebrated partition congruences: p(5n + 4) ≡ 0(mod 5), p(7n + 5) ≡ 0(mod 7) and, p(11n + 6) ≡ 0(mod 11), where p(n) denotes the number of partitions of n. We interpret these congruences using the notions of rank and crank, and vector partitions—concepts developed by Dyson, Atkin, Swinnerton-Dyer, Andrews, and Garvan via elegant combinatorial arguments and q-series identities.
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References
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