Partitions, Durfee Symbols and the Atkin-Garvan Moments of Ranks

George Andrews, Penn State

Abstract: In 1944, Freeman Dyson defined the rank of a partition with the object of providing a combinatorial interpretation of the Ramanujan conguences for the partition function, p(n). Dyson's discoveries and conjectures have led to an extensive field of research with exciting discoveries by Atkin, Garvan, Swinnerton-Dyer, Ono, Bringmann and Mahlburg and others. In this talk, we consider moments that Atkin and Garvan associated with the ranks defined by Dyson. We shall reveal the objects they enumerate and shall discuss implications and possibilities.

About the speaker: Professor Andrews is Evan Pugh Professor of Mathematics at Penn State. His research centers on the theory of partitions and related areas. He has a long-term interest in the work of S. Ramanujan, whose last notebook he unearthed in 1976. He is now collaborating with Bruce Berndt on a series of volumes explicating the brilliant and sometimes enigmatic ideas in this notebook. He was elected to the American Academy of Arts and Sciences in 1997 and the National Academy of Sciences in 2003.