The average number of linear extensions of a partial order (Q1906134)

\textit{D. J. Kleitman} and \textit{B. L. Rothschild} [Trans. Am. Math. Soc. 205, 205-220 (1975; Zbl 0302.05007)] gave an asymptotic formula for the number of partial orders in an \(n\) element set. The authors give a shorter proof of this result and extend it to count linear extensions of a given partial order. Thus asymptotic formulas are obtained for the average number of linear extensions of an \(n\) element partial order and for the number of suborders of an \(n\) element linear order. Some graph theory technique is used in the proofs.