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

Abstract

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 $${\nu(q)}$$ and relates the even part of $${\nu(q)}$$ to the partitions initially considered. We also consider a surprisingly simple combinatorial relationship between the cranks and the ranks of the partition of n.