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Determinantal inequalities for the partition function

  • Dennis X.Q. Jia (a1) and Larry X.W. Wang (a1)

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 $n\ges 222$ ,
$$(p(n)^2-p(n-1)p(n+1))^2-(p(n-1)^2-p(n-2)p(n))(p(n+1)^2-p(n)p(n+2))>0.$$

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