Stirling Numbers

$$S(n+1,k)=kS(n,k)+S(n,k-1)$$ $$S(n,0)==\{\begin{array}{ll}0& \text{if }n\ne 0\\ 1& \text{if }n=0\end{array}$$

$\tilde{s}(n,k)$ is the number of permutations of the set $[n]$ whose cycle representation hast exactly k cycles

Falling Factorial