Abstract
We study the partition rank function N(r, t; n), the number of partitions with rank congruent to r modulo t. We first show that it is monotonic in n above a given bound, and then show that it equidistributed as \(n \rightarrow \infty \). Using this result we prove a conjecture of Hou and Jagadeeson on the convexity of N(r, t; n).
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References
Bessenrodt, C., Ono, K.: Maximal multiplicative properties of partitions. Ann. Comb. 20(1), 59–64 (2016)
Bringmann, K.: Asymptotics for rank partition functions. Trans. Am. Math. Soc. 361(7), 3483–3500 (2009)
Bringmann, K., Ono, K.: Dyson’s ranks and Maass forms. Ann. Math. 419–449 (2010)
Bringmann, K., Folsom, A., Ono, K., Rolen, L.: Harmonic Maass forms and mock modular forms: theory and applications. Am. Math. Soc. 64 (2017)
Chan, S.H., Mao, R.: Inequalities for ranks of partitions and the first moment of ranks and cranks of partitions. Adv. Math. 258, 414–437 (2014)
Dyson, F.: Some guesses in the theory of partitions. Eureka (Cambridge) 8, 10–15 (1944)
Garvan, F.G.: New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11. Trans. Am. Math. Soc. 305(1), 47–77 (1988)
Hardy, G., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. Lond. Math. Soc. 2(1), 75–115 (1918)
Hou, E., Jagadeesan, M.: Dyson’s partition ranks and their multiplicative extensions. Ramanujan J. 45(3), 817–839 (2018)
Ingham, A.: A Tauberian theorem for partitions. Ann. Math. 1075–1090 (1941)
Jennings-Shaffer, C., Reihill, D.: Asymptotic formulas related to the \(M_2\)-rank of partitions without repeated odd parts. Ramanujan J. (2018)
Mumford, D., Nori, M., Norman, P.: Tata Lectures on Theta III, vol. 43. Springer, New York (2007)
Ramanujan, S.: Congruence properties of partitions. Math. Z. 9, 147–153 (1919)
Zwegers, S.: Mock theta functions, Ph.D. Thesis, Universiteit Utrecht (2002)
Zwegers, S.: Multivariable Appell functions and nonholomorphic Jacobi forms. Res. Math. Sci. 6(1), 16 (2019)
Acknowledgements
The author would like to thank Kathrin Bringmann for helpful comments on previous versions of the paper, as well as Chris Jennings-Shaffer for useful conversations. The author would also like to thank the referee for many helpful comments.
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Males, J. Asymptotic equidistribution and convexity for partition ranks. Ramanujan J 54, 397–413 (2021). https://doi.org/10.1007/s11139-019-00202-8
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DOI: https://doi.org/10.1007/s11139-019-00202-8