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.
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Dedicated to the memory of Richard Askey, a great friend and mentor.
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The author was supported in part by a grant from the Simon’s Foundation (#318714). A preliminary version of this paper was presented October 1, 2020 at the Number Theory Seminar, St. Petersburg State University and Euler International Mathematical Institute, Russia. It was also presented October 3, 2020 at the Special Session on q-Series and Related Areas in Combinatorics and Number Theory, A.M.S. Fall Eastern Sectional Meeting.
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Garvan, F.G. A new approach to the Dyson rank conjectures. Ramanujan J 61, 545–566 (2023). https://doi.org/10.1007/s11139-021-00530-8
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DOI: https://doi.org/10.1007/s11139-021-00530-8