Let $S(m, n)$ denote the number of onto functions from a set with m elements to a set with $n$ elements. Show that $S(m, n)$ satisfies the recurrence relation
$$S(m, n) = n^{m} − \sum_{k=1 }^{n−1} C(n, k)S(m, k)$$ whenever $m \geq n$ and $n > 1,$ with the initial condition $S(m, 1) = 1.$