Abstract
Let k be a positive integer and m be an integer. Garvan’s k-rank \(N_k(n,m)\) is the number of partitions of n into at least \((k-1)\) successive Durfee squares with k-rank equal to m. In this paper we give some asymptotics for \(N_k(n,m)\) with \(|m|\ge \sqrt{n}\) as \(n\rightarrow \infty .\) As a corollary, we give a more complete answer for the Dyson’s crank distribution conjecture. We also establish some asymptotic formulas for finite differences of \(N_k(n,m)\) with respect to m with \(m\gg \sqrt{n}\log n.\)
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dyson, F.J.: Some guesses in the theory of partitions. Eureka 8, 10–15 (1944)
Andrews, G.E., Garvan, F.G.: Dyson’s crank of a partition. Bull. Am. Math. Soc. (N.S.) 18(2), 167–171 (1988)
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)
Dyson, F.J.: Mappings and symmetries of partitions. J. Comb. Theory A 51(2), 169–180 (1989)
Mao, R.: Asymptotic inequalities for \(k\)-ranks and their cumulation functions. J. Math. Anal. Appl. 409(2), 729–741 (2014)
Bringmann, K., Manschot, J.: Asymptotic formulas for coefficients of inverse theta functions. Commun. Number Theory Phys. 7(3), 497–513 (2013)
Kim, B., Kim, E., Seo, J.: Asymptotics for \(q\)-expansions involving partial theta functions. Discrete Math. 338(2), 180–189 (2015)
Bringmann, K., Dousse, J.: On Dyson’s crank conjecture and the uniform asymptotic behavior of certain inverse theta functions. Trans. Am. Math. Soc. 368(5), 3141–3155 (2016)
Garvan, F.G.: Generalizations of Dyson’s rank and non-Rogers–Ramanujan partitions. Manuscr. Math. 84(3–4), 343–359 (1994)
Dousse, J., Mertens, M.H.: Asymptotic formulae for partition ranks. Acta Arith. 168(1), 83–100 (2015)
Parry, D., Rhoades, R.C.: On Dyson’s crank distribution conjecture and its generalizations. Proc. Am. Math. Soc. 145(1), 101–108 (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)
Hardy, G.H., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. Lond. Math. Soc. 2(17), 75–115 (1918)
Odlyzko, A.M.: Differences of the partition function. Acta Arith. 49(3), 237–254 (1988)
Acknowledgements
The author would like to thank the anonymous referees for their very helpful comments and suggestions. The author also thank Professor Zhi-Guo Liu for his consistent encouragement.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhou, N.H. On the distribution of rank and crank statistics for integer partitions. Res. number theory 5, 18 (2019). https://doi.org/10.1007/s40993-019-0156-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40993-019-0156-z