Abstract
The paper begins with a study of a couple of classes of partitions in which each even part is smaller than each odd. In one class, a Dyson-type crank exists to explain a mod 5 congruence. The second part of the paper treats the arithmetic and combinatorial properties of the third order mock theta function \({\nu(q)}\) and relates the even part of \({\nu(q)}\) to the partitions initially considered. We also consider a surprisingly simple combinatorial relationship between the cranks and the ranks of the partition of n.
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Andrews, G.E. Integer Partitions with Even Parts Below Odd Parts and the Mock Theta Functions. Ann. Comb. 22, 433–445 (2018). https://doi.org/10.1007/s00026-018-0398-9
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DOI: https://doi.org/10.1007/s00026-018-0398-9