PARTITION CONGRUENCES AND DYSON’S RANK

In this article the rank of a partition of an integer is a certain integer associated with the partition. The term has first introduced by freeman Dyson in a paper published in Eureka in 1944. In 1944, F.S. Dyson discussed his conjectures related to the partitions empirically some Ramanujan’s famous partition congruences. In 1921, S. Ramanujan proved his famous partition congruences: The number of partitions of numbers 5n+4, 7n+5 and 11n +6 are divisible by 5, 7 and 11 respectively in another way. In 1944, Dyson defined the relations related to the rank of partitions. These are later proved by Atkin and Swinnerton-Dyer in 1954. The proofs are analytic relying heavily on the properties of modular functions. This paper shows how to generate the generating functions for N(1,n),N(?1,n),N(2,n),N(m,n),N(0,5,n),N(1,5,n)... In this paper, we show how to prove the Dyson’s conjectures with rank of partitions.