In generalized birthday paradox problem
The probability of getting $k$ unique values from $[0, n)$ when choosing $m$ times is given by:
$$P(V = k) = \binom{n}{k}\displaystyle\sum_{i=0}^k (-1)^i \binom{k}{i} \left(\frac{k-i}{n}\right)^m $$
where $V$ is a random variable giving the number of unique outcomes and $\binom{\cdot}{\cdot}$ is the binomial coefficient.
Sampling from such distribution can be easily achieved by sampling from discrete uniform distribution and then counting the number of unique values. Is there any better way from random generation from this distribution? I call the direct simulation inefficient because it needs generating each time $m$ samples from the discrete uniform distribution.