Linear Extension Diameter of Downset Lattices of 2-Dimensional Posets

TL;DR

This paper investigates the maximum distance between linear extensions in downset lattices of 2-dimensional posets, providing formulas, characterizations, and polynomial-time algorithms for computing this diameter.

Contribution

It introduces a formula for the linear extension diameter of Boolean lattices and characterizes diametral pairs for downset lattices of 2D posets, with efficient computation methods.

Findings

Formula for the linear extension diameter of Boolean lattices

Characterization of diametral pairs in downset lattices of 2D posets

Polynomial-time algorithm for computing the diameter

Abstract

The linear extension diameter of a finite poset P is the maximum distance between a pair of linear extensions of P, where the distance between two linear extensions is the number of pairs of elements of P appearing in different orders in the two linear extensions. We prove a formula for the linear extension diameter of the Boolean Lattice and characterize the diametral pairs of linear extensions. For the more general case of a downset lattice D_P of a 2-dimensional poset P, we characterize the diametral pairs of linear extensions of D_P and show how to compute the linear extension diameter of D_P in time polynomial in |P|.