# Inequality with Stirling's numbers of the second kind

Prove inequality:

$S(n,k-1)\cdot S(n,k+1) \le (S(n,k))^2$

for $n,k\in\mathbb{N}$, where $S(n,k)$ is the Stirling number of the second kind.

I think induction is the last resort. I don't know if it will work but I rather want to avoid it.

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A google search with the keywords "Stirling numbers" and "log-concave" gives several hits, like ism.ac.jp/editsec/aism/pdf/040_4_0693.pdf and mathdl.maa.org/images/upload_library/22/Ford/… . –  darij grinberg Mar 9 '12 at 18:02
Asked and answered at m.se, math.stackexchange.com/questions/118029/… - voting to close. –  Gerry Myerson Mar 10 '12 at 4:46
@darij grinberg I don't understand why in ism.ac.jp/editsec/aism/pdf/040_4_0693.pdf the inductive step is on $n$. Why does it work.. –  xan Mar 10 '12 at 13:17