Let $(P, \le)$ be a poset on $n$ elements $x_1\dots x_n$. A total order $<$ on the same set is said to be a linear extension of $\le$ if $(\forall i,j)\quad x_i \le x_j \rightarrow x_i < x_j$.
The problem of counting the number of linear extensions of a given poset is known to be $#P-complete$: this is proved in Brightwell, Graham R.; Winkler, Peter (1991), "Counting linear extensions", Order 8 (3): 225–242.
In the same paper some bounds are given to estimate this number. These bounds are improved in Kahn, J.; Kim, J. H. (1992), "Entropy and sorting", Proocedings of the 24th Annual ACM Symposium on Theory of Computing: 178-187
Were these bounds improved again? What are the best known bounds for this problem?
Béla Bollobás, Graham Brightwell, and Alexander Sidorenko, Geometrical techniques for estimating numbers of linear extensions, European J. Combin. 20 (1999), no. 5, 329–335, MR1702372 (2000j:05007) improves on one of the results in the Kahn and Kim paper, though I'm not sure it improves on that particular bound.