It's hard to count the number of posets with a given number of elements; Sloane's A001035 has it only up to 18. Are asymptotic results known?
My actual interest is in transitive relations, Sloane's A006905, but it's not hard to convert between these. I would accept an answer giving either.
 Götz Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences 7 (2004).
 R. P. Stanley, Enumerative Combinatorics, 1986.