The bounds for the number of linear extensions via chain and antichain coverings

Abstract

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 $e(\mathcal{P})\geqslant \prod b_i!$.