Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-23T23:03:16.684Z Has data issue: false hasContentIssue false
  • English
  • Français

Determinantal inequalities for the partition function

Part of: Polynomials and matrices Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  29 January 2019

Dennis X.Q. Jia  and
Larry X.W. Wang
Dennis X.Q. Jia
Affiliation:
Center for Combinatorics, Nankai University, Tianjin300071, P. R. China ([email protected]; [email protected])
Larry X.W. Wang
Affiliation:
Center for Combinatorics, Nankai University, Tianjin300071, P. R. China ([email protected]; [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let p(n) denote the partition function. In this paper, we will prove that for $n\ges 222$,

$$\left| {\matrix{ {p(n)} & {p(n + 1)} & {p(n + 2)} \cr {p(n-1)} & {p(n)} & {p(n + 1)} \cr {p(n-2)} & {p(n-1)} & {p(n)} \cr } } \right| > 0.{\rm }$$
As a corollary, we deduce that p(n) satisfies the double Turán inequalities, that is, for $n\ges 222$,
$$(p(n)^2-p(n-1)p(n+1))^2-(p(n-1)^2-p(n-2)p(n))(p(n+1)^2-p(n)p(n+2))>0.$$

Keywords

Partition functionlog-concavitydeterminantthe Hardy-Ramanujan-Rademacher formuladouble Turán inequality

MSC classification

Primary: 05A20: Combinatorial inequalities
Secondary: 11P82: Analytic theory of partitions 11C20: Matrices, determinants
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1451 - 1466
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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.)

References

1Aissen, M., Edrei, A., Schoenberg, I. J. and Whitney, A. M.. On the generating functions of totally positive sequence. Proc. Nat. Acad. Sci. 37 (1951), 303–307.10.1073/pnas.37.5.303CrossRefGoogle Scholar
2Brenti, F.. Unimodal, log-concave, and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc. 81 (1989).Google Scholar
3Chen, W. Y. C.. Recent developments on log-concavity and q-log-concavity of combinatorial polynomials. In FPSAC 2010 Conf. Talk Slides, http://www.billchen.org/talks/2010-FPSAC.pdf (2010).Google Scholar
4Chen, W. Y. C., Jia, D. X. Q. and Wang, L. X. W.. Higher order Turán inequalities for the partition function. Trans. Amer. Math. Soc. (to appear), http://doi.org/10.1090/tran/7707.CrossRefGoogle Scholar
5Chen, W. Y. C., Wang, L. X. W. and Xie, G. Y. B.. Finite differences of the logarithm of the partition function. Math. Comp. 85 (2016), 825–847.10.1090/mcom/2999CrossRefGoogle Scholar
6Craven, T. and Csordas, G.. Jensen polynomials and the Turán and Laguerre inequalities. Pacific J. Math. 136 (1989), 241–260.10.2140/pjm.1989.136.241CrossRefGoogle Scholar
7Craven, T. and Csordas, G.. Karlin's conjecture and a question of Pólya. Rocky Moutain J. Math. 35 (2005), 61–82.CrossRefGoogle Scholar
8Csordas, G. and Dimitrov, D. K.. Conjectures and theorems in the theory of entire functions. Numer. Alg. 25 (2000), 109–122.CrossRefGoogle Scholar
9DeSalvo, S. and Pak, I.. Log-concavity of the partition function. Ramanujan J. 38 (2015), 61–73.10.1007/s11139-014-9599-yCrossRefGoogle Scholar
10Hou, Q. H. and Zhang, Z. R.. r-log-concavity of partition functions. Ramanujan J. (2018). https://doi.org/10.1007/s11139-017-9975-5.CrossRefGoogle Scholar
11Karlin, S.. Total positivity, vol. I (California: Standford University Press, 1968).Google Scholar
12Laguerre, E.. Oeuvres, vol. 1 (Paris: Gaauthier-Villars, 1989).Google Scholar
13Levin, B.Ja.. Distribution of zeros of entire functions, revised edn, Translations of Mathematical Monographs, vol. 5 (Providence, R.I.: American Mathematical Society, 1980).Google Scholar
14Pólya, G. and Schur, J.. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math. 144 (1914), 89–113.Google Scholar
15Rahman, Q. I. and Schmeieer, G.. Analytic theroy of polynomials (Oxford: Oxford University Press, 2002).Google Scholar