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

TL;DR

This paper derives formulas for counting downsets and antichains in finite posets, and uses these to explicitly determine the number of extensions with fixed minimal points, revealing deep combinatorial properties and bounds.

Contribution

It introduces exponential functions of finite posets to count extensions with fixed minimal points, providing explicit formulas, bounds, and asymptotic relations.

Findings

Formulas for downsets and antichains in finite posets

Explicit enumeration of posets with fixed minimal points

Bounds and asymptotic behavior of counting functions

Abstract

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.