The Combinatorics of Differentiation

Let S_1, S_2, ... be a sequence of finite sets, and suppose we are asked to find the sequence of cardinalities s[1], s[2], .... We are usually satisfied to find a closed-form expression for the a-generating function F_S(z)=∑_(n ≥ 0) s[n]a[n]z^n, where a[n] is a fixed positive causal sequence. But extracting s[n] from F_S (z) is often itself a challenging problem, because of the unnavoidable link to calculus s[n] = (a[n])/(n!)D^n[F(z)]_z = 0. In this paper we will consider the case a[n] = 1/(n!), (exponential generating functions), and find many links between combinatorics and calculus.