A correlational inequality for linear extensions of a poset

Abstract

Suppose 1, 2, and 3 are pairwise incomparable points in a poset onn≥3 points. LetN (ijk) be the number of linear extensions of the poset in whichi precedesj andj precedesk. Define λ by

$$\lambda = \frac{{N(213)N(312)}}{{\left[ {N(123) + N(132)} \right]\left[ {N(231) + N(321)} \right]}}$$

Two applications of the Ahlswede-Daykin evaluation theorem for distributive lattices are used to prove that λ⩽(n−1)²/(n+1)² for oddn, and λ⩽(n−2)/(n+2) for evenn. Simple examples show that these bounds are best-possible.

Shepp (Annals of Probability, 1982) proved thatP(12)⩽P(12/13), the so-calledxyz inequality, whereP(ij) is the probability thati precedesj in a randomly chosen linear extension of the poset, thus settling a conjecture of I. Rival and B. Sands. The preceding bounds on λ yield a simple proof ofP(12)<P(12/13), which had also been conjectured by Rival and Sands.