In 1990, new statistics on partitions (called cranks) were found which combinatorially prove Ramanujan's congruences for the partition function modulo 5, 7, 11 and 25. The methods are extended to find cranks for Ramanujan's partition congruence modulo 49. A more explicit form of the crank is given for the modulo 25 congruence.
Let p(n) denote the number of unrestricted partitions of n. The congruences referred to in the title are p(5n + 4), pCln + 5) and p(lln + 6) = 0 (mod 5, 7 and 11, respectively). Dyson conjectured and Atkin and Swinnerton-Dyer proved combinatorial results which imply the congruences mod 5 and 7. These are in terms of the rank of partitions. Dyson also conjectured the existence of a "crank" which would likewise imply the congruence mod 11. In this paper we give a crank which not only gives a combinatorial interpretation of the congruence mod 11 but also gives new combinatorial interpretations of the congruences mod 5 and 7. However, our crank is not quite what Dyson asked for; it is in terms of certain restricted triples of partitions, rather than in terms of ordinary partitions alone.Our results and those of Dyson, Atkin and Swinnerton-Dyer are closely related to two unproved identities that appear in Ramanujan's "lost" notebook. We prove the first identity and show how the second is equivalent to the main theorem in Atkin and Swinnerton-Dyer's paper. We note that all of Dyson's conjectures mod 5 are encapsulated in this second identity. We give a number of relations for the crank of vector partitions mod 5 and 7, as well as some new inequalities for the rank of ordinary partitions mod 5 and 7. Our methods are elementary relying for the most part on classical identities of Euler and Jacobi.
Abstract. New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.
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