 Solution

For a surjection, the nonempty fibres $f^{-1}(b)$ form a partition of a set $A$ into exactly $n$ blocks. Choose these unlabelled blocks in $S(m,n)$ ways, then assign the $n$ distinct elements of $B$ to them in $n!$ ways. ConsequentlyIt is zero when $m<n$. With $S(0,0)=1$ and $S(m,0)=0$ for $m>0$, the same formula also handles an empty set codomain.