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 $n>657$.

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.