I am going through Phil Hanlon's paper and on page 127, right after the first paragraph, "It is well known that.." which boils down to the following identity:

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

where $c(\sigma)$ is the number of cycles (of any length) in a permutation.

I suspect this is not a difficult thing to prove, but I have not been able to find any literature that deals directly with this identity. Any help?

From the context, I think $ \beta $ is positive and strictly less than 1.