CRANKS FOR PARTITIONS WITH BOUNDED LARGEST PART

For 80 years, Dyson's rank has been known as the partition statis-tic that witnesses the first two of Ramanujan's celebrated congruences for the ordinary partition function. In this paper, we show that Dyson's rank actu-ally witnesses families of partition congruences modulo every prime P. This comes from an in-depth study of when a "multiplicity-based statistic" is a crank witnessing congruences for the function p`n, m, which enumerates par-titions of n with parts of size at most m. We also show that as the modulus P increases, there is an ever-growing collection of distinct multiplicity-based cranks witnessing these same families.