In 1944, Freeman Dyson initiated an important subject in the
theory of partitions
by discovering a simple statistic called the rank. He
conjectured that the rank
provided a combinatorial explanation for
Ramanujan's congruences for the partition
function modulo 5 and 7. In 1954, Atkin and Swinnerton-Dyer proved
Dyson's
conjectures by establishing generating functions for certain rank
differences in
arithmetic progressions. In this talk, we discuss analogous results
for a generalization
of partitions called overpartitions.
This is joint work with Jeremy Lovejoy.