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edited Apr 13, 2017 at 12:19

It seems that the correct formula is actually

$$ \prod_{i=0}^{n-1}(\beta+i) = \sum_{\sigma \in S_n}\beta^{c(\sigma)} $$

And this has a quite nice proof over on math.SE here here.

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created Jul 24, 2016 at 20:08

It seems that the correct formula is actually

$$ \prod_{i=0}^{n-1}(\beta+i) = \sum_{\sigma \in S_n}\beta^{c(\sigma)} $$

And this has a quite nice proof over on math.SE here.