  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.
