Abstract: Let p(n) denote the number of unrestricted partitions of n. The congruences referred to in the title are p(5n+4), p(7n+5) and p(11n+6) 0 (mod 5, 7 and 11, respectively). Dyson conjectured and Atkin and Swinnerton-Dyer proved 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 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 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 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.