The solution for the number of distributions leaving none of the $n$ cells empty (with unlike cells and $r$ unlike objects) is given by
$$A(r,n)=\sum_{\nu=0}^{n-1}(-1)^{\nu}\binom{n}{\nu}(n-\nu)^{r}$$ (although I have seen the same expression summing from $\nu=0$ to $n$).
By the way, if anyone knows which one is correct, please let me know. However, that's not the question.
I have read that this expression provides a solution to a famous problem. I'd like to know where can I find more information about this solution and which famous problem solves. I already tried An Introduction to Combinatorial Analysis by John Riordan and Certain distributions of unlike objects into cells by Morton Abramson but their treatment of this particular expression is very brief.
Thanks in advance.