This problem is the same as number of ways to partition n into exactly m parts.
The recurrence given in Wikipedia has
p(n,k) = the number of partitions of n using only natural numbers ≥ k
How to find no of partitions of n which has exactly k non-zero parts? Is there a recurrence relation to solve this?
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 $n > 0$, $$ p(n,k) = \sum_{t \leq \lfloor n/k \rfloor} p(n-kt,k+1). $$
