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Determinantal inequalities for the partition function

Part of: Additive number theory; partitions Polynomials and matrices 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 (dennisjxq@mail.nankai.edu.cn; wsw82@nankai.edu.cn)
Larry X.W. Wang
Affiliation:
Center for Combinatorics, Nankai University, Tianjin300071, P. R. China (dennisjxq@mail.nankai.edu.cn; wsw82@nankai.edu.cn)
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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

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