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.


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

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

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.