Partitions with short sequences and mock theta functions

P. A. MacMahon was the first to examine integer partitions in which consecutive integers were not allowed as parts. Such partitions may be described as having sequences of length 1. Recently it was shown that partitions containing no sequences of consecutive integers of length k are of interest in seemingly unrelated problems concerning threshold growth models. The object now is to develop the subject intrinsically to both provide deeper understanding of the theory and application of partitions and reveal the surprising role of Ramanujan's mock theta functions.