Partitions with short sequences
and Ramanujan's mock theta functions

In a recent paper "Integrals, Partitions and Cellular Automata" in the Transactions of the American Mathematical Society, Holroyd, Liggett and Romik evaluated an intriguing definite integral and applied it to a variety of probability models. The application to integer partitions concerned partitions in which no sequence of consecutive integers of length k appears (k=2,3,...). The authors note that in one instance a proof of their result can also be based on a little known partition theorem of P.A.MacMahon. Our object in this talk will be to introduce these ideas and to develop the study of such partitions from a purely combinatorial, q-series point of view. Surprisingly the one of Ramanujan's mysterious mock theta functions arises.