Is the sequence of partition numbers log-concave? Let $p(n)$ denote the number of partitions of a positive integer $n$. It seems to me that we have for all $n>25$
$$
p(n)^2>p(n-1)p(n+1).
$$
In other words, the sequence $(p(n))_{n\in \mathbb{N}}$ is log-concave, or satisfies $PF_2$, with
$$
\det \begin{pmatrix} p(n) & p(n+1) \cr p(n-1) & p(n) \end{pmatrix}>0
$$
for $n>25$. Is this true ? I could not find a reference in the literature so far. On the other hand, the partition function is really studied a lot.
So it seems likely that this is known. 
Similarly, property $PF_3$, with the corresponding $3\times 3$ determinant, seems to hold for all $n>221$, too, and also
$PF_4$ for all $n>657$. 
The question is also motivated from the study of Betti numbers for nilpotent Lie algebras, in particular filiform nilpotent Lie algebras.
 A: This paper from a J. Janoski at Clemson seems to indicate that despite the fact that partitions have been studied half-to-death, the log concavity is still somewhat open (AND the asymptotic way of doing it is the only way known). Note that a related unimodality theorem of Szekeres (for partitions into $k$ parts) is only proved using asymptotics, and not a bijective correspondence, so the "book proofs" of both facts still elude us.
A: The first two terms of the Hardy-Ramanujan formula give
$$p(n) = \frac{1}{4 \sqrt{3} n} \exp(\pi \sqrt{2n/3}) + O \left(\exp(\pi  \sqrt{n/6} ) \right)$$
so
$$\log p(n) = \pi \sqrt{2/3} \sqrt{n} - \log n - \log (4 \sqrt{3}) + O(\exp(-\pi  \sqrt{n/6} ) ).$$
So
$$\log p(n+2) - 2 \log p(n+1) + \log p(n) = $$
$$ \pi \sqrt{2/3} \left( \sqrt{n+2} - 2\sqrt{n+1} + \sqrt{n} \right) - \left( \log(n+2) - 2 \log(n+1) + \log n \right) + O(\exp(-\pi  \sqrt{n/6} ) )$$
$$= \left[ \left( \frac{- \pi \sqrt{2/3}}{4} \right) n^{-3/2} + O(n^{-5/2}) \right] + O(n^{-2}) +  O(\exp(-\pi  \sqrt{n/6} ) ).$$
So this quantity is negative for $n$ sufficiently large.
The larger determinants seem harder; there is probably a smarter way to do this.

With the help of Mathematica, I set $q(n) = a \exp(c \sqrt{n})/n$ and computed that
$$\det \begin{pmatrix} q(n) & q(n+1) & q(n+2) \\ q(n-1) & q(n) & q(n+1) \\ q(n-2) & q(n-1) & q(n) \end{pmatrix} = q(n)^3 \left( \frac{c^3}{32 n^{9/2}} + O(n^{-10/2}) \right).$$
The error in approximating $p(n)$ by $q(n)$ (for $a = 1/(4 \sqrt{3})$ and $c = \pi \sqrt{2/3}$) will be exponentially smaller than $n^{-9/2}$, so the $3 \times 3$ determinant is positive for $n$ large. 
The $4 \times 4$ determinant vanishes to order at least $n^{-12/2}$, and I gave up waiting for the computation to finish when I asked for more terms.
A: The statement referenced by Igor Rivin http://www.math.clemson.edu/~janoski/ResearchStatement.pdf uses the phrase

Computationally looking at p(n) we see that for n ≥ 26 the partition function is log-concave [2].

I had seen this reference before probably about the same time this research statement was first released, and I am skeptical for two reasons.


*

*The phrasing "Computationally..." would seem to indicate some type of calculation.  This cannot involve a computer since it would have to hold for all n larger than 26, and I am not aware of any simplification that allows one to only consider a finite number of cases.  It would have been helpful to at least expound on the type of computations involved.

*I checked for the promised reference, and indeed I found it on the CV of the author, http://www.math.clemson.edu/~janoski/VitaTex.pdf, but it refers to the quote below.  I did a quick google search and I could find no reference or anything pointing to a publication.

Brian Bowers, Neil Calkin, Kerry Gannon, Janine E. Janoski, Katie Joes, Anna Kirkpatrick, The Log Concavity of the Partition Function, (in preparation)


*Asymptotics will not provide the answer here, since n sufficiently large doesn't hold up unless you can provide a concrete n and test everything less than it, and I don't believe the Hardy-Ramanujan asymptotic expansion yields any guaranteed error estimates.

*It may be possible to use DH Lehmer's estimates to obtain a proof.  In two papers (1937 and 1939) he investigated the coefficients of both the Hardy-Ramanujan asymptotic expansion and the Hardy-Ramanujan-Rademacher expansion.  He provided guaranteed error bounds on the remainder terms in the asymptotic expansions so that, for example, his Theorem 13 says that for n>600, only $2/3 \sqrt n$ terms of the Hardy-Ramanujan asymptotic series are needed to estimate p(n) to the nearest integer.
At present, I don't believe the matter is completely settled, despite the overwhelming computational evidence.
UPDATE 11-1-13: 
Igor Pak and I have just uploaded a preprint to the ArXiv: http://arxiv.org/abs/1310.7982 .  In it we prove the log-concavity of the partition numbers for all $n>25$, and Section 6.3 addresses Janoski's thesis.
UPDATE 11-23-15:
Igor and I were recently informed of work by Jean-Louis Nicolas which also contains a proof of the log-concavity of the partition numbers:
Sur les entiers N pour lesquels il y a beaucoup de groupes abéliens d’ordre N,
Annales de l’institut Fourier, tome 28, no 4 (1978), p. 1-16.
http://www.numdam.org/item?id=AIF_1978__28_4_1_0
