Abstract
We study the Dyson rank function N(r, 3; n), the number of partitions of n with rank \(\equiv r \pmod 3\). We investigate the convexity of these functions. We extend N(r, 3; n) multiplicatively to the set of partitions, and we determine the maximum value when taken over all partitions of size n.
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Notes
Dyson’s rank does not explain Ramanujan’s congruence modulo 11.
This follows immediately from considering conjugations of Ferrers diagrams.
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Acknowledgements
The authors would like to thank the NSF and the Emory REU (especially Professor Ono) for their support of our research.
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Funding was provided by National Science Foundation (Grant Nos. DMS-1557960 and DMS-1557960).
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Hou, E., Jagadeesan, M. Dyson’s partition ranks and their multiplicative extensions. Ramanujan J 45, 817–839 (2018). https://doi.org/10.1007/s11139-016-9881-2
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DOI: https://doi.org/10.1007/s11139-016-9881-2