I'm doing some research in a different area of math and I need the following answered:
Let $s_{n,k}$ denote the unsigned Stirling numbers of the first kind, $0 \leq k \leq n$. Does there exist a constant $C > 0$ such that for all $k,n$, $$s_{n,k} < C^n(n-k)!$$
I hope the answer is no.
Any pointers towards the relevant literature would be appreciated too.