We know that the number of $s$-partition of an $n$-element set is given by the Stirling number of the second kind. However, is there any result on the number of $s$-partition such that every component has at least $k$ elements, where $k$ is any integer?
In particular, it would be great if we can get some value such that the number that we are seeking is of the same order, or equivalently, find some upper and lower bound as a function of $n,s,k$.
Thanks in advance.
One can write down its exponential generating function: it's the coefficient of $x^n/n!$ in $$\frac1{s!}\left(\sum_{r=k}^\infty\frac{x^r}{r!}\right)^s.$$