The average number of linear extensions of a partial order

Kleitman and Rothschild (Trans. Amer. Math. Soc. 205 (1975), 205–220) gave an asymptotic formula for the number of partial orders with ground-set [n]. We give a shorter proof of their result and extend it to count the number of pairs (P, ≺), where P is a partial order on [n] and ≺ is a linear extension of P. This gives us an asymptotic formula for (a) the average number of linear extensions of an n-element partial order and (b) the number of suborders of an n-element linear order