Linear Extensions and Comparable Pairs in Partial Orders

© 2017 Springer Science+Business Media B.V. We study the number of linear extensions of a partial order with a given proportion of comparable pairs of elements, and estimate the maximum and minimum possible numbers. We also consider a random interval partial order on n elements, which has close to a third of the pairs comparable with high probability: we show that the number of linear extensions is n! 2−Θ(n) with high probability.