Number theoretic properties of generating functions related to Dyson’s rank for partitions into distinct parts. (English) Zbl 1216.11093
Summary: Let \(Q(n)\) denote the number of partitions of \(n\) into distinct parts. We show that Dyson’s rank provides a combinatorial interpretation of the well-known fact that \(Q(n)\) is almost always divisible by 4. This interpretation gives rise to a new false theta function identity that reveals surprising analytic properties of one of Ramanujan’s mock theta functions, which in turn gives generating functions for values of certain Dirichlet \(L\)-functions at nonpositive integers.
MSC:
| 11P82 | Analytic theory of partitions |
| 11P83 | Partitions; congruences and congruential restrictions |
Online Encyclopedia of Integer Sequences:
Triangle read by rows: T(n,k) = number of partitions into distinct parts having rank k, 0<=k<n.References:
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