A combinatorial approach towards Ramanujan’s partition congruences modulo 5, 7 and 11 in terms of its corresponding cranks

Abstract The primary purpose of this article is to provide an explicit interpretation of Ramanujan’s Partition Congruences, namely, p(5n + 4) ≡ 0 (mod 5), p(7n + 5) ≡ 0 (mod 7) and, p(11n + 6) ≡ 0 (mod 11), p(n) being the partition function corresponding to any positive integer n, in terms of their corresponding Cranks using various combinatorial arguments implemented by Dyson, Atkin, Swinnerton-Dyer and later by Andrews and, Garvan to justify all the necessary concepts pertaining to this topic. In addition to introducing readers to the Theory of Partitions, as well as defining formally the notion of q-Series and Ramanujan’s Theta Function, we shall further deduce a rough estimate for p(n), a priori applying the so called Jacobi Triple Product Identity and Euler Pentagonal Theorem, and subsequently, providing a thorough explanation for the special case, when n ≡ 4 (mod 5). Important to mention that, a whole section in this article has been dedicated towards gaining an detailed understanding of Rank of a partition and Crank of a Vector Partition, both introduced by Dyson, due to its significance and relevance to our study. Furthermore, in the later half of the text, we shall indeed prove our main result with the help of an important property of the partition function M(m, j, n) corresponding to any positive integer n with crank congruent to m modulo j. Exclusively, in the final section, rigorous derivations have been provided individually for each case to facilitate the motivated readers of this paper with a better understanding of this subject. Moreover, adequate references have been included, considering the broader aspect of research in this area. 2020 MSC: Primary 11-02, 11P81, 11P82, 11P83, 11P84. Secondary 11B75, 05A16, 05A17, 05A30.