From Dyson's rank to false theta functions: the story of a partition congruence identity

Let Q(n) denote the number of partitions of n into distinct parts. It is known that Q(n) is almost always (for n in a dense subset of the positive integers) divisible by any fixed power of 2. We give a new combinatorial explanation for the fact that Q(n) is nearly always divisible by 2 and 4, and investigate the analytic properties of the corresponding generating functions that arise from the combinatorial proof.