journal-of-discrete-mathematical-sciences-and-cryptography Journal of Discrete Mathematical Sciences and Cryptography 2169-0065 0972-0529 Taru Publications 10.47974/JDMSC-2416 A combinatorial approach towards Ramanujan’s partition congruences modulo 5, 7 and 11 in terms of its corresponding cranks De Subham Department of Mathematics, Hauz Khas, Indian Institute of Technology Delhi, New Delhi, Delhi, 110016, India 29 7 2685 2698 30 03 2026

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.

 Partition Rank Crank Vector partition q-Series Ramanujan’s theta function Jacobi-triple product identity Euler’s pentagonal theorem Successive rank access open retracted no