A note on linear extensions and incomparable pairs

Abstract Given 2 ⩽ m ⩽ n , let P ( m , n ) be the family of partially ordered sets on {1, 2, …, n } in which {1, …, m } is an antichain. We identify each ordered set in P ( m , n ) that maximizes the proportion of all linear extensions in which 1 is above 2. The proof features a series of applications of the strict xyz inequality.