Let $n$ be a non-zero natural integer. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$.
Establish that, for any real $x$,
$$\prod_{i=0}^{n-1}(x+i) = \sum_{k=1}^{n} s(n,k) x^k.$$