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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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  • $\begingroup$ Suppose that $p=k=3$ and $a=1$. Your formula yields $[z^3]\big((1+z)^3\big)=1$, but the correct answer is clearly $3!=6$, since the items are distinct. $\endgroup$ – Brian M. Scott Nov 8 '11 at 0:48
  • $\begingroup$ Thanks, I am regarding exponential generating functions, and should have made that clear at the coefficient extraction stage. $\endgroup$ – Phira Nov 8 '11 at 7:24

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