  We establish formulas for the number of all downsets (or equivalently, of all
antichains) of a finite poset P. Then, using these numbers, we determine
recursively and explicitly the number of all posets having a fixed set of
minimal points and inducing the poset P on the non- minimal points. It turns
out that these counting functions are closely related to a collection of
downset numbers of certain subposets. Since any function of that kind is an
exponential sum (with the number of minimal points as exponent), we call it the
exponential function of the poset. Some linear equations, divisibility
relations, upper and lower bounds, and asymptotical equalities for the counting
functions are deduced. A list of all such exponential functions for posets with
up to five points concludes the paper.
