No of ways in which n indistinguishable items can be placed in m indistinguishable boxes
The number of partitions of $n$ with exactly $k$ parts is the coefficient of $x^n y^k$ in the generating series
$$ \prod_{l=1}^\infty \frac{1}{1-x^ly}. $$
Asymptotically, this is probably $\Theta_k(n^{k-1})$ (i.e. the constant depends on $k$).

The number of partitions of $n$ in which each part is at least $k$ is
$$ \prod_{l=k}^\infty \frac{1}{1-x^l}. $$
Asymptotically, my guess is that it's $\Theta_k(p(n))$, where $p(n)$ is the number of all partitions of $n$.

We can write a recurrence relation for $p(n,k)$: $p(0,k) = 1$ and for $n &gt; 0$,
$$ p(n,k) = \sum_{t \leq \lfloor n/k \rfloor} p(n-kt,k+1). $$