I need to prove such a formula:
$ \left\{ \begin{matrix} n \\ k-1 \end{matrix} \right\} \cdot \left\{ \begin{matrix} n \\ k+1 \end{matrix} \right\} \le \left\{ \begin{matrix} n \\ k \end{matrix} \right\}^{2} \\ $ Where {} are Stirling numbers of the second kind.
(number of ways to partition a set of n objects into k non-empty subsets).
I tried to figure out some combinatorial proof, but failed.
I'd be grateful for any help.