Let $p(n)$ denote the number of partitions of a positive integer $n$. It seems to me that we have for all $n>25$
$$
p(n)^2>p(n1)p(n+1).
$$
In other words, the sequence $(p(n))_{n\in \mathbb{N}}$ is logconcave, or satisfies $PF_2$, with
$$
\det \begin{pmatrix} p(n) & p(n+1) \cr p(n1) & p(n) \end{pmatrix}>0
$$
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.



The first two terms of the HardyRamanujan 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 $$\det \begin{pmatrix} q(n) & q(n+1) & q(n+2) \\ q(n1) & q(n) & q(n+1) \\ q(n2) & q(n1) & q(n) \end{pmatrix} = q(n)^3 \left( \frac{c^3}{32 n^{9/2}} + O(n^{10/2}) \right).$$ 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. 


The statement referenced by Igor Rivin http://www.math.clemson.edu/~janoski/ResearchStatement.pdf uses the phrase
I had seen this reference before probably about the same time this research statement was first released, and I am skeptical for two reasons.
At present, I don't believe the matter is completely settled, despite the overwhelming computational evidence. UPDATE 11113: Igor Pak and I have just uploaded a preprint to the ArXiv: http://arxiv.org/abs/1310.7982 . In it we prove the logconcavity of the partition numbers for all $n>25$, and Section 6.3 addresses Janoski's thesis. UPDATE 112315: Igor and I were recently informed of work by JeanLouis Nicolas which also contains a proof of the logconcavity 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. 116. 


This paper from a J. Janoski at Clemson seems to indicate that despite the fact that partitions have been studied halftodeath, 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. 

