Abstract
We give asymptotic expansions for the moments of the \(M_2\)-rank generating function and for the \(M_2\)-rank generating function at roots of unity. For this we apply the Hardy–Ramanujan circle method extended to mock modular forms. Our formulas for the \(M_2\)-rank at roots of unity lead to asymptotics for certain combinations of N2(r, m, n) (the number of partitions without repeated odd parts of n with \(M_2\)-rank congruent to r modulo m). This allows us to deduce inequalities among certain combinations of N2(r, m, n). In particular, we resolve a few conjectured inequalities of Mao.
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The authors thank Kathrin Bringmann for suggesting this project and for useful comments and discussions.
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Jennings-Shaffer, C., Reihill, D. Asymptotic formulas related to the \(M_2\)-rank of partitions without repeated odd parts. Ramanujan J 52, 175–242 (2020). https://doi.org/10.1007/s11139-019-00147-y
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DOI: https://doi.org/10.1007/s11139-019-00147-y
Keywords
- Integer partitions
- Partition ranks
- Rank differences
- Asymptotics
- Circle method
- Rank inequalities
- \(M_2\)-rank
- Harmonic Maass forms
- Modular forms
- Mock modular forms
- Mock theta functions