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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.

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    Combinatorial proof was rather my personal guess. I just noticed that huge amount of Stirling-number-related formulas have clear and easy combinatorial proof, that's all.2012-03-09

0 Answers 0