Hostname: page-component-669899f699-ggqkh Total loading time: 0 Render date: 2025-04-24T18:30:07.459Z Has data issue: false hasContentIssue false
  • English
  • Français

INEQUALITIES AND UNIFORM ASYMPTOTIC FORMULAE FOR SPT-CRANK OF PARTITIONS

Part of: Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  13 September 2024

YUAN CHEN  and
NIAN HONG ZHOU Open the ORCID record for NIAN HONG ZHOU [Opens in a new window]
YUAN CHEN
Affiliation:
School of Mathematics and Statistics, Guangxi Normal University, Guilin, Guangxi, PR China e-mail: [email protected]
NIAN HONG ZHOU*
Affiliation:
School of Mathematics and Statistics, Guangxi Normal University, Guilin, Guangxi, PR China
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish some inequalities that arise from truncating Lerch sums and derive uniform asymptotic formulae for the spt-crank of ordinary partitions. The uniform asymptotic formulae improve upon a result of Mao [‘Asymptotic formulas for spt-crank of partitions’, J. Math. Anal. Appl. 460(1) (2018), 121–139].

Keywords

integer partitionsspt-crankuniform asymptotics

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 111 , Issue 1 , February 2025 , pp. 56 - 67
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Nian Hong Zhou was partially supported by the National Natural Science Foundation of China (No. 12301423) and the Key Laboratory of Mathematical Model and Application (Guangxi Normal University), Education Department of Guangxi Zhuang Autonomous Region.

References

Andrews, G. E., ‘The number of smallest parts in the partitions of $n$ ’, J. reine angew. Math. 624 (2008), 133–142.Google Scholar
Andrews, G. E., Chan, S. H. and Kim, B., ‘The odd moments of ranks and cranks’, J. Combin. Theory Ser. A 120(1) (2013), 77–91.CrossRefGoogle Scholar
Andrews, G. E., Garvan, F. G. and Liang, J., ‘Combinatorial interpretations of congruences for the spt-function’, Ramanujan J. 29(1–3) (2012), 321–338.CrossRefGoogle Scholar
Chen, W. Y. C., Ji, K. Q. and Zang, W. J. T., ‘The spt-crank for ordinary partitions’, J. reine angew. Math. 711 (2016), 231–249.CrossRefGoogle Scholar
Hardy, G. H. and Ramanujan, S., ‘Asymptotic formulae in combinatory analysis’, Proc. Lond. Math. Soc. (2) 17 (1918), 75–115.CrossRefGoogle Scholar
Liu, Z.-G. and Zhou, N. H., ‘Uniform asymptotic formulas for the Fourier coefficients of the inverse of theta functions’, Ramanujan J. 57(3) (2022), 1085–1123.CrossRefGoogle Scholar
Mao, R., ‘Asymptotic formulas for spt-crank of partitions’, J. Math. Anal. Appl. 460(1) (2018), 121–139.CrossRefGoogle Scholar
Ramanujan, S., ‘Congruence properties of partitions’, Math. Z. 9(1–2) (1921), 147–153.CrossRefGoogle Scholar