Asymptotic enumeration of partially ordered sets

Abstract

The author define the entropy function S( ρ)=Lim_{n- ∞} 2n⁻²lnN (n, ρ ), where N(n, ρ ) is the number of distinct partial order relations which may be defined on a set of n elements such that a fraction ρ of the possible n(n−1)/2 pairs are comparable.We derive upper bounds to S(ρ) to show that S(ρ)<(l/2) In 2 if ρ>.699.