Linear Extensions and Comparable Pairs in Partial Orders

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 $n! \, 2^{-\Theta(n)}$ with high probability.