Ramanujan's congruences and Dyson's crank

The achievement of Karl Mahlburg in this issue of PNAS (1) adds a lustrous chapter to a unique mathematical object: the crank. In 1944, the crank was first hinted at by Freeman Dyson (2), then an undergraduate at Cambridge University. He had written an article, titled Some Guesses in the Theory of Partitions, for Eureka, the undergraduate mathematics journal of Cambridge. Dyson discovered the many conjectures in this article by attempting to find a combinatorial explanation of Ramanujan's famous congruences for p(n), the number of partitions of n. The three simplest of Ramanujan's congruences assert that: In Dyson's article, he defines the rank of a partition to be the largest part of the partition minus the number of parts. For example, the rank of the partition (of 22) 5 + 5 + 3 + 3 + 3 + 2 + 1 is 5 - 7 = -2. Dyson conjectured (and later Atkin and Swinnerton-Dyer proved, see ref. …