Integer Partitions with Even Parts Below Odd Parts and the Mock Theta Functions

The paper begins with a study of a couple of classes of partitions in which each even part is smaller than each odd. In one class, a Dyson-type crank exists to explain a mod 5 congruence. The second part of the paper treats the arithmetic and combinatorial properties of the third order mock theta function ν(q) and relates the even part of ν(q) to the partitions initially considered. We also consider a surprisingly simple combinatorial relationship between the cranks and the ranks of the partition of n.