Abstract
Let j, n be even positive integers, and let \(\overline{p}_j(n)\) denote the number of partitions with BG-rank j, and \(\overline{p}_j(a,b;n)\) to be the number of partitions with BG-rank j and 2-quotient rank congruent to \(a \ \, \left( \mathrm {mod} \, b \right) \). We give asymptotics for both statistics, and show that \(\overline{p}_j(a,b;n)\) is asymptotically equidistributed over the congruence classes modulo b. We also show that each of \(\overline{p}_j(n)\) and \(\overline{p}_j(a,b;n)\) asymptotically satisfy all higher-order Turán inequalities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Berkovich and F. Garvan, On the Andrews-Stanley refinement of Ramanujan’s partition congruence modulo 5 and generalizations, Trans. Amer. Math. Soc., 358 (2006), no. 2, 703–726.
C. Bessenrodt and K. Ono, Maximal multiplicative properties of partitions, Ann. Comb. 20 (2016), no. 1, 59–64.
K. Bringmann, On the explicit construction of higher deformations of partition statistics, Duke Math. J., 144 (2008), no. 2, 195 – 233.
K. Bringmann and J. Dousse, On Dyson’s crank conjecture and the uniform asymptotic behavior of certain inverse theta functions, Trans. Amer. Math. Soc. 368 (2016), no. 5, 3141–3155.
K. Bringmann,and K. Mahlburg, Asymptotic inequalities for positive crank and rank moments, Trans. Amer. Math. Soc., 366 (2014), no. 2, 1073–1094.
K. Bringmann, W. Craig, J. Males, and K. Ono, Distributions on partitions arising from Hilbert schemes and hook lengths, Forum Math. Sigma, 10, E49.
K. Bringmann, C. Jennings-Shaffer, K. Mahlburg, and R. Rhoades, Peak positions of strongly unimodal sequences, Trans. Amer. Math. Soc. 372 (2019), no. 10, 7087–7109.
G. Cesana, W. Craig, and J. Males, Asymptotic equidistribution for partition statistics and topological invariants, preprint, https://arxiv.org/abs/2111.13766.
H. Cohen and F. Stromberg, Modular forms: a classical approach, vol 179 of Graduate Studies in Mathematics. American Mathematical Society, 2017.
W. Craig and A. Pun, A note on the higher order Turán inequalities for \(k\)-regular partitions, Res. Number Theory 7 (2021), no. 1, Paper No. 5, 7 pp.
M. Dawsey and R. Masri, Effective bounds for the Andrews spt-function, Forum Math. 31 (2019), no. 3, 743–767.
S. DeSalvo and I. Pak, Log-concavity of the partition function, Ramanujan J. 38 (2015), 1, 61–73.
D. Dimitrov, Higher order Turán inequalities, Proc. Amer. Math. Soc., 126 (1998), no. 7, 2033–2037.
M. Griffin, K. Ono, L. Rolen, and D. Zagier, Jensen polynomials for the Riemann zeta function and other sequences, Proc. Natl. Acad. Sci. USA, 116 (2019), 23, 11103–11110.
G. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. Ser. 2 17 (1918), 75–115.
E. Hou and M. Jagadeesan, Dyson’s partition ranks and their multiplicative extensions, Ramanujan J. 45 (2018), no. 3, 817–839.
H. Larson and I. Wagner, Hyperbolicity of the partition Jensen polynomials, Res. Number Theory, 5 (2019), Paper No. 19.
D. Littlewood, Modular representations of symmetric groups, Proc. Roy. Soc. London Ser. A, 209 (1951), 333–353.
J. Males, Asymptotic equidistribution and convexity for partition ranks, Ramanujan J. 54 (2021), no. 2, 397–413.
H. Ngo and R. Rhoades, Integer Partitions, Probabilities and Quantum Modular Forms, Res. Math. Sci. 4 (2017), Paper No. 17, 36 pp.
F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds., NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.1.6 of 2022-06-30.
K. Ono, S. Pujahari, and L. Rolen, Turán inequalities for the plane partition function, preprint, https://arxiv.org/abs/2201.01352.
H. Rademacher, A convergent series for the partition function \(p(n)\), PNAS February 1, 1937 23 (2), 78–84.
J. Schur and G. Pólya, Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math., 144 (1914), 89–113.
G. Szegö, On an inequality of P. Turán concerning Legendre polynomials, Bull. Amer. Math. Soc., 54 (1948), 401–405.
I. Wagner, On a new class of Laguerre-Pólya type functions with applications in number theory, preprint, arXiv: 2108.01827.
E. Wright, Stacks. II, Quart. J. Math. Oxford Ser. 22 (1971), no. 2, 107–116.
Acknowledgements
The authors thank the Manitoba eXperimental Mathematics Laboratory for their support of the project.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no competing interests to declare that are relevant to the content of this article.
Additional information
Communicated by Jang Soo Kim.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The research of the Joshua Males conducted for this paper is supported by the Pacific Institute for the Mathematical Sciences (PIMS). The research and findings may not reflect those of the Institute.
Andrew Baker: Undergraduate author at the University of Manitoba.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Baker, A., Males, J. Asymptotics, Turán Inequalities, and the Distribution of the BG-Rank and 2-Quotient Rank of Partitions. Ann. Comb. 27, 769–780 (2023). https://doi.org/10.1007/s00026-022-00612-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00026-022-00612-4