On the 1/3–2/3 Conjecture

Abstract

Let (P, ≤) be a finite poset (partially ordered set), where P has cardinality n. Consider linear extensions of P as permutations x₁x₂⋯x_n in one-line notation. For distinct elements x, y ∈ P, we define ℙ(x ≺ y) to be the proportion of linear extensions of P in which x comes before y. For $0\leq \alpha \leq \frac {1}{2}$, we say (x, y) is an α-balanced pair if α ≤ ℙ(x ≺ y) ≤ 1 − α. The 1/3–2/3 Conjecture states that every finite partially ordered set which is not a chain has a 1/3-balanced pair. We make progress on this conjecture by showing that it holds for certain families of posets. These include lattices such as the Boolean, set partition, and subspace lattices; partial orders that arise from a Young diagram; and some partial orders of dimension 2. We also consider various posets which satisfy the stronger condition of having a 1/2-balanced pair. For example, this happens when the poset has an automorphism with a cycle of length 2. Various questions for future research are posed.