Rank of a partition

[1] It was presented in the context of a study of certain congruence properties of the partition function discovered by the Indian mathematical genius Srinivasa Ramanujan.

A different concept, sharing the same name, is used in combinatorics, where the rank is taken to be the size of the Durfee square of the partition.

[1] Srinivasa Ramanujan in a paper published in 1919 proved the following congruences involving the partition function p(n):[2] In commenting on this result, Dyson noted that " .

although we can prove that the partitions of 5n + 4 can be divided into five equally numerous subclasses, it is unsatisfactory to receive from the proofs no concrete idea of how the division is to be made.

[1] Dyson introduced the idea of rank of a partition to accomplish the task he set for himself.

[3] The following tables show how the partitions of the integers 4 (5 × n + 4 with n = 0) and 9 (5 × n + 4 with n = 1 ) get divided into five equally numerous subclasses.