  We describe a Markov-Chain-Monte-Carlo algorithm which can be used to
generate naturally labeled n-element posets at random with a probability
distribution of one's choice. Implementing this algorithm for the uniform
distribution, we explore the approach to the asymptotic regime in which almost
every poset takes on the three-layer structure described by Kleitman and
Rothschild (KR). By tracking the n-dependence of several order-invariants,
among them the height of the poset, we observe an oscillatory behavior which is
very unlike a monotonic approach to the KR regime. Only around n=40 or so does
this "finite size dance" appear to give way to a gradual crossover to
asymptopia which lasts until n=85, the largest n we have simulated.
