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Let ( P , ≤ ) $(\mathcal {P},\leqslant )$ be a finite poset. Define the numbers a 1 , a 2 ,… (respectively, c 1 , c 2 ,…) so that a 1 + … + a k (respectively, c 1 + … + c k ) is the maximal number of elements of P $\mathcal {P}$ which may be covered by k antichains (respectively, k chains.) Then the number e ( P ) $e(\mathcal {P})$ of linear extensions of poset P $\mathcal {P}$ is not less than ∏ a i ! $\prod a_{i}!$ and not more than n ! / ∏ c i ! $n!/\prod c_{i}!$ . A corollary: if P $\mathcal {P}$ is partitioned onto disjoint antichains of sizes b 1 , b 2 ,…, then e ( P ) ≥ ∏ b i ! $e(\mathcal {P})\geqslant \prod b_{i}!$ .

The Bounds for the Number of Linear Extensions Via Chain and Antichain Coverings