According to this Wikipedia article, there is an explicit formula for Stirling numbers of the first kind:
$$\sum_{1\le i_1<i_2<\cdots<i_{k-1}<n} \frac{(n-1)!}{i_1i_2\cdots i_{k-1}} =s(n,k)$$
What is the proof of this? I understand if it is too long to post here, so a reference direction would be fine.