Given a natural number $n$, I want to construct a (finite) poset $P_n$ such that $P_n$ has exactly $n$ linear extensions.
This can always be done, for instance taking $P_n$ to be a chain of length $n-1$ plus a single element that is incomparable to all the others.
However, I would like each $P_n$ to have a "small" number of elements. More specifically, I would like to know if there such a sequence $(P_n)$ such that, for all $n$, $P_n$ has exactly $n$ linear extensions and $|P_n| = O(\log(n))$, where $|P_n|$ is the number of elements of $P_n$. Is there a construction for this, or a way to justify that it cannot be done?
More generally, is anything known about the sequence that maps $n$ to the size of the smallest poset with exactly $n$ linear extensions?