  Let $(\mathcal{P},\leqslant)$ be a finite poset. Define the numbers
$a_1,a_2,\ldots$ (respectively, $c_1,c_2,\ldots$) so that $a_1+\ldots+a_k$
(respectively, $c_1+\ldots+c_k$) is the maximal number of elements of
$\mathcal{P}$ which may be covered by $k$ antichains (respectively, $k$
chains.) Then the number $e(\mathcal{P})$ of linear extensions of poset
$\mathcal{P}$ is not less than $\prod a_i!$ and not more than $n!/\prod c_i!$.
A corollary: if $\mathcal{P}$ is partitioned onto disjoint antichains of size
$b_1,b_2, \ldots$, then $e(\mathcal{P})\geqslant \prod b_i!$.
