Some Properties of Partitions

WE denote by p{n) the number of unrestricted partitions of a positive integer n. Ramanujan discovered, and later proved, three striking arithmetical properties of p{n), namely: #(571+4) = 0 (mod 5), (1.1) p{ln+5) = = 0(mod7), (1.2) p{Un+6) = 0 (mod 11). (1.3) All existing proofs of these results appeal to the theory of generating functions, and provide no method of actually separating the partitions concerned into q equal classes {q = 5, 7, or 11). Dyson (1) discovered empirically a remarkable combinatorial method of dividing the partitions of 5w+4 and ln-\\-5 into 5 and 7 equal classes respectively. Defining the rank of a partition as the largest part minus the number of parts, he divided the partitions of any number into 5 classes according to their ranks modulo 5. For numbers of the form 5w+4, these 5 classes are all equal, while for numbers of other forms some but not all of the classes are equal; similar