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

Let ( P , ≤ ) 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 which may be covered by k antichains (respectively, k chains.) Then the number e ( P ) of linear extensions of poset P is not less than ∏ a i ! and not more than n ! / ∏ c i ! . A corollary: if P is partitioned onto disjoint antichains of sizes b 1 , b 2 ,…, then e ( P ) ≥ ∏ b i ! .