I am trying to find some asymptotic expression for the unsigned stirling numbers of the first kind.
Lets denote them by $|s(n,k)|$, and suppose that $k$ is fixed.
So far I have tried using the fact that $$|s(n,k)| = \sum\{ i_1i_2\ldots i_{n-k} : 1 \leq i_1 < i_2 < \ldots < i_{n-k} \leq n-1 \}$$ And trying to start from $$ \frac{1}{(n-k)!}\left( \sum_{i=1}^{n-1}i \right)^{n-k} = \frac{1}{(n-k)!} {n \choose 2}^{n-k}$$ Which contains all products of the form $\{ i_1i_2\ldots i_{n-k} : i_j \in [0,n-1] \}$ divided by that factor of $\frac{1}{(n-k)!}$, so I was trying to sepparate the products of distinct terms from the rest, and trying to see that the rest should be "small". But I got stuck from here.
So, can somebody give me any tips, or help me with this approach, or should I look for asymptotics by using the recurrence relation, the generating function or maybe some other technique?
Thanks,