In the first part of the question I was asked to find the exponential generating function for $s_{n,r}$, the number of ways to distribute $r$ distinct objects into $n$ (a fixed constant) distinct boxes with no empty box.
I determined the generating function to be $(e^x -1 )^n$. The problem then asks to determine $s_{n,r}$. I'm not sure how this is any different from the first part of the question, but I thought maybe it was asking for a coefficient of $x^r / r!$ How do I do this?
- 3$\begingroup$This question is similar to:About the Stirling number of the second kind. If you believe it’s different, pleaseedit the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem.$\endgroup$DroneBetter– DroneBetter2025-11-29 06:35:28 +00:00Commented5 hours ago
1 Answer1
Employ the following two facts:
$$(e^x-1)^n=\sum_{k=0}^n{n\choose k}e^{kx}(-1)^{n-k},\qquad e^{kx}=\sum_{r\ge0}\frac{k^r}{r!}x^r.$$
The first is from the binomial theorem and second is from the Taylor expansion of $\exp$.
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