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