Cranks for partitions with bounded largest part

Abstract

For 80 years, Dyson’s rank has been known as the partition statistic that witnesses the first two of Ramanujan’s celebrated congruences for the ordinary partition function. In this paper, we show that Dyson’s rank actually witnesses families of partition congruences modulo every prime ℓ \ell . This comes from an in-depth study of when a “multiplicity-based statistic” is a crank witnessing congruences for the function p ( n , m ) p\big (n,m\big ) , which enumerates partitions of n n with parts of size at most m m . We also show that as the modulus ℓ \ell increases, there is an ever-growing collection of distinct multiplicity-based cranks witnessing these same families.