Linear Extension Diameter of Downset Lattices of 2-Dimensional Posets

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 Boolean Lattices and characterize all pairs of linear extensions attaining the maximum distance. These results can be extended to all downset lattices of 2-dimensional posets.