Enumerative combinatorics
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Let S1, S2, ... 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 $F_S(z) = \sum_{n \geq 0} s[n]a[n] { z^n}$, where a[n] is a fixed positive causal sequence. But extracting s[n] from FS(z) is often itself a challenging problem, because of the unnavoidable link to calculus $s[n] ={a[n] \over n!} D^n[F(z)]_{z=0}$. In this paper we will consider the case a[n] = 1/n!, (exponential generating functions), and find many links between combinatorics and calculus.