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How to distribute $k$ distinct items into $p$ distinct groups with each groups receiving $a (=k-n)$ prizes at most ?

This is my attempt to generalize the constraints of my earlier question and based on the discussion/comments in this answer.

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You want a list of $p$ sets of sets of size at most $a$.

The exponential generating function for sets of size at most $a$ is $ 1+\frac{z}{1!}+\dots + \frac{z^a}{a!}$

So, finally, you want to extract the coefficient of $\frac{z^k}{k!}$ from

$ \left(1+\frac{z}{1!}+\dots+\frac{z^a}{a!}\right)^p.$

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    Thanks, I am regarding exponential generating functions, and should have made that clear at the coefficient extraction stage.2011-11-08