New combinatorial interpretations of Ramanujan’s partition congruences mod 5,7 and 11

Let p (n) p (n) denote the number of unrestricted partitions of n n. The congruences referred to in the title are p (5 n + 4) p (5n + 4), p (7 n + 5) p (7n + 5) and p (11 n + 6) ≡ 0 p (11n + 6) 0 (mod 5 5, 7 7 and 11 11, respectively). Dyson conjectured and Atkin and Swinnerton-Dyer proved combinatorial results which imply the congruences mod 5 5 and 7 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 11. In this paper we give a crank which not only gives a combinatorial interpretation of the congruence mod 11 11 but also gives new combinatorial interpretations of the congruences mod 5 5 and 7 7. However, our crank is not quite what Dyson asked for; it is in terms of certain restricted tri