A new approach to the Dyson rank conjectures

Abstract

In 1944 Dyson defined the rank of a partition as the largest part minus the number of parts, and conjectured that the residue of the rank mod 5 divides the partitions of $$5n+4$$ into five classes of equal size. This gave a combinatorial explanation of Ramanujan’s famous partition congruence mod 5. He made an analogous conjecture for the rank mod 7 and the partitions of $$7n+5$$ . In 1954 Atkin and Swinnerton-Dyer proved Dyson’s rank conjectures by constructing several Lambert-series identities basically using the theory of elliptic functions. In 2016 the author gave another proof using the theory of weak harmonic Maass forms. In this paper we describe a new and more elementary approach using Hecke–Rogers series.