Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T00:55:59.428Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE DISTRIBUTION OF THE RANK STATISTIC FOR STRONGLY CONCAVE COMPOSITIONS

Part of: Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  13 February 2019

NIAN HONG ZHOU Open the ORCID record for NIAN HONG ZHOU [Opens in a new window]
NIAN HONG ZHOU*
Affiliation:
School of Mathematical Sciences, East China Normal University, Shanghai 200241, PR China email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A strongly concave composition of $n$ is an integer partition with strictly decreasing and then increasing parts. In this paper we give a uniform asymptotic formula for the rank statistic of a strongly concave composition introduced by Andrews et al. [‘Modularity of the concave composition generating function’, Algebra Number Theory7(9) (2013), 2103–2139].

Keywords

concave compositionpartitionsrankasymptotics

MSC classification

Primary: 11P82: Analytic theory of partitions
Secondary: 05A16: Asymptotic enumeration 05A17: Partitions of integers
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 2 , October 2019 , pp. 230 - 238
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

Footnotes

This research was supported by the National Science Foundation of China (Grant No. 11571114).

References

Andrews, G. E., ‘Concave and convex compositions’, Ramanujan J. 31(1–2) (2013), 67–82.Google Scholar
Andrews, G. E. and Garvan, F. G., ‘Dyson’s crank of a partition’, Bull. Amer. Math. Soc. (N.S.) 18(2) (1988), 167–171.Google Scholar
Andrews, G. E., Rhoades, R. C. and Zwegers, S. P., ‘Modularity of the concave composition generating function’, Algebra Number Theory 7(9) (2013), 2103–2139.Google Scholar
Bringmann, K. and Dousse, J., ‘On Dyson’s crank conjecture and the uniform asymptotic behavior of certain inverse theta functions’, Trans. Amer. Math. Soc. 368(5) (2016), 3141–3155.Google Scholar
Bringmann, K. and Mahlburg, K., ‘An extension of the Hardy–Ramanujan circle method and applications to partitions without sequences’, Amer. J. Math. 133(4) (2011), 1151–1178.10.1353/ajm.2011.0025Google Scholar
Bringmann, K. and Manschot, J., ‘Asymptotic formulas for coefficients of inverse theta functions’, Commun. Number Theory Phys. 7(3) (2013), 497–513.Google Scholar
Byungchan, K., Eunmi, K. and Jeehyeon, S., ‘Asymptotics for q-expansions involving partial theta functions’, Discrete Math. 338(2) (2015), 180–189.Google Scholar
Dousse, J. and Mertens, M. H., ‘Asymptotic formulae for partition ranks’, Acta Arith. 168(1) (2015), 83–100.Google Scholar
Dyson, F. J., ‘Some guesses in the theory of partitions’, Eureka 8 (1944), 10–15.Google Scholar
Dyson, F. J., ‘Mappings and symmetries of partitions’, J. Combin. Theory Ser. A 51(2) (1989), 169–180.Google Scholar
Garvan, F. G., ‘New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11’, Trans. Amer. Math. Soc. 305(1) (1988), 47–77.Google Scholar
Hardy, G. H. and Ramanujan, S., ‘Asymptotic formulae in combinatory analysis’, Proc. Lond. Math. Soc. (2) 17 (1918), 75–115.Google Scholar
Mao, R., ‘Asymptotic inequalities for k-ranks and their cumulation functions’, J. Math. Anal. Appl. 409(2) (2014), 729–741.Google Scholar
Parry, D. and Rhoades, R. C., ‘On Dyson’s crank distribution conjecture and its generalizations’, Proc. Amer. Math. Soc. 145(1) (2017), 101–108.Google Scholar