Partition congruences and the Andrews-Garvan-Dyson crank

Abstract

In 1944, Freeman Dyson conjectured the existence of a ''crank'' function for partitions that would provide a combinatorial proof of Ramanujan's congruence modulo 11. Forty years later, Andrews and Garvan successfully found such a function and proved the celebrated result that the crank simultaneously ''explains'' the three Ramanujan congruences modulo 5, 7, and 11. This note announces the proof of a conjecture of Ono, which essentially asserts that the elusive crank satisfies exactly the same types of general congruences as the partition function.

Dyson’s ranks and Maass forms

“…(3) Theorem 1.5 is in sharp contrast to Mahlburg's recent result on the AndrewsGarvan-Dyson crank (see [23]). For example, his results imply that congruences modulo Q j exist for all the crank partition functions with modulus t D Q.…”

“…In this direction, we shall employ a method first used by the second author in [24] in his work on p.n/. We show that Dyson's rank partition functions satisfy congruences of Ramanujan type, a result which nicely complements the recent paper [23] by Mahlburg on the AndrewsGarvan-Dyson crank. THEOREM 1.5.…”

Partitions, Durfee symbols, and the Atkin–Garvan moments of ranks

“…Most recently K. Mahlburg [23] has revealed that the crank explains in a grand and glorious manner the infinite panoply of partition congruences discovered and proved by K. Ono et al [2], [24], [25].…”

The meaning of Ramanujan now and for the future

“…However, it should be stressed that there are numerous recent spectacular discoveries by Ono, Bringmann, Mahlburg, Zwegers and others related to Ramanujan's mock theta functions [20], [21], [36], [56].…”