Predators and Cranks

Predators and cranks.

Cranks and T -cores

New statistics on partitions (called cranks) are defined which combinatorially prove Ramanujan’s congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given for the equinumerous crank classes. The cranks are closely related to the t-core of a partition. Using q-series, some explicit formulas are given for the number of partitions which are t-cores. Some related...

Inequalities Between Ranks and Cranks

Higher moments of the partition rank and crank statistics have been studied for their connections to combinatorial objects such as Durfee symbols, as well as for their connections to harmonic Maass forms. This paper proves the first several cases of (and strengthens) a conjecture due to Garvan, which states that the moments of the crank function are always larger than the moments of the crank f...

Cranks – Really, the Final Problem

A survey of Ramanujan’s work on cranks in his lost notebook is given. We give evidence that Ramanujan was concentrating on cranks when he died, that is to say, the final problem on which Ramanujan worked was cranks – not mock theta functions.

The odd moments of ranks and cranks

have motivated much research. Here, p(n) denotes the number of partitions of n. In particular, toward a combinatorial explanation of the above congruences many partition statistics have been studied. Among them, the rank suggested by F. Dyson [6] and the crank suggested by the first author and F.G. Garvan [2] have proven successful and their own properties have been extensively studied. Here, t...