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?