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For integers $n\ge k\ge0$, let $p(n,k)$ denote the number of ways to write $n$ as a sum of $k$ positive integers (repetition allowed). For example, $p(6,3)=3$ since $$6=1+1+4=1+2+3=2+2+2.$$

QUESTION. Is it true that the inequality $$p(n,k)^2>p(n,k-1)p(n,k+1)\tag{1}$$ holds whenever $1\le k\le n-26$?

I conjecture that the answer is positive, this implies that for any integer $n\ge27$ the integer sequence $(p(n,k))_{1\le k\le n-26}$ is log-concave. For $n=27,28,\ldots,100$, I have verified the inequality $(1)$ for all $k=1,\ldots,n-26$.

Your comments are welcome! If you know that the question is not new, please provide me a reference.

Edit. S. DeSalvo and I. Pak [Ramanujan J. 38 (2015), 61-73] proved the inequality $p(n)^2>p(n-1)p(n+1)$ for any integer $n\ge 26$. Thus, in view of Fedor Petrov's helpful comments, for any integer $m\ge26$ and $n\ge 2m+2$ we have $$p(n,n-m)^2>p(n,n-m+1)p(n,n-m-1),\ \text{i.e.,}\ p(m)^2>p(m-1)p(m+1).$$ This shows that $(1)$ holds if $n/2+1\le k\le n-26$. So, it remains to prove that $(1)$ is valid when $1\le k\le\min\{(n+1)/2,n-26\}$.

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    $\begingroup$ it worth mention that for fixed $c$ and $n\geqslant 2c$, we have $p(n,n-c)=p(c)$, thus this strengthens the log-concavity of $p(n)$ $\endgroup$
    – Fedor Petrov
    Commented Sep 28 at 9:31
  • $\begingroup$ It is very decent to cite related results, if you are aware of them. $\endgroup$
    – Ilya Bogdanov
    Commented Sep 28 at 10:02
  • $\begingroup$ @Fedor Petrov. Thank you for pointing out that. $\endgroup$
    – Zhi-Wei Sun
    Commented Sep 28 at 13:42
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    $\begingroup$ An asymptotic estimate for $p(n,k)$ was obtained by G. Szekeres, Some asymptotic formulae in the theory of partitions (II), Quart. J. Math. Oxford 4 (2), 96--111, that implies the unimodality of the sequence $p(n,1), p(n,2), \dots, p(n,n)$. Does this estimate or some refinement of it imply log-concavity? $\endgroup$
    – Richard Stanley
    Commented Sep 28 at 15:42
  • $\begingroup$ @Richard Stanley Thank you for informing me the paper of Szekeres on the asymptotic behavior of $p(n,k)$. This paper might be helpful. $\endgroup$
    – Zhi-Wei Sun
    Commented Sep 28 at 22:56

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