Exponential functions of finite posets and the number of extensions with a fixed set of minimal points

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 ar that kind is an exponential sum (with the number of minimal points as exponents), we call it the exponential function of the poset. Some linear equalities, divisibility relations, upper and lower bounds. A list of all such exponential functions for posets with up to five points concludes the paper.