On Dyson’s crank distribution conjecture and its generalizations

Abstract:

Bringmann and Dousse recently established a conjecture of Dyson dealing with the limiting asymptotics of the Andrews-Garvan crank statistic for integer partitions. This note presents a direct “sieving” technique to establish this conjecture. The technique readily yields the analogous result for Dyson’s partition rank and all of Garvan’s $k$-rank statistics.