k-Run Overpartitions and Mock Theta Functions

In this paper we introduce k-run overpartitions as natural analogs to partitions without k-sequences, which were first defined and studied by Holroyd, Liggett, and Romik. Following their work as well as that of Andrews, we prove a number of results for k-run overpartitions, beginning with a double summation q-hypergeometric series representation for the generating functions. In the special case of 1-run overpartitions we further relate the generating function to one of Ramanujan’s mock theta functions. Finally, we describe the relationship between k-run overpartitions and certain sequences of random events, and use probabilistic estimates in order to determine the asymptotic growth behavior of the number of k-run overpartitions of size n.