The Stirling number of the second kind, $S(n,k)$, is defined to be the number of ways one can partition an $n$-element set into exactly $k$ subsets. The sum over the values for $k$ from 1 to $n$ becomes the Bell number. One asymptotic approximation for $S(n,k)$ is $\frac{k^n}{k!}$.
I just wonder, what is the asymptotic approximations of partitioning an $n$-element set into at most $k$ subsets. That is,
$\sum_{k=1}^{p} S(n,k)$,
the sum over the values for $k$ from 1 to $p$, where $p$ is a constant? Is there any tighter approximation than $\frac{p^n}{(p-1)!}$?