The following question has arisen tangentially in relation to the analysis of the complexity of an algorithm in my Master's thesis, but my background in combinatorics is limited, and I'm having trouble finding a clear answer:
Let $(P,<)$ be a finite poset. A subset $M \subseteq P$ is a maximal chain if $(M,<)$ is a chain and there is no $N \subseteq P$ such that $M \subsetneq N$ and $(N,<)$ is a chain. The height of $(P,<)$ is the length of the longest maximal chain in $P$.
Now, suppose we fix $n∈ℕ$, and choose a poset $(P,<)$ uniformly at random from the set of posets with n elements.
What is a good upper bound on the expected value of the height of $(P,<)$?