Noncrossing Partitions From Hull Configurations

TL;DR

This paper explores the structure of noncrossing partition lattices derived from hull configurations, unifying cases where points lie on a line segment or polygon boundary, and reveals their combinatorial properties.

Contribution

It introduces hull configurations and proves their associated lattices are unions of Boolean subposets with potential symmetric chain decompositions.

Findings

Lattices from hull configurations are unions of Boolean subposets.

Under certain conditions, these lattices admit symmetric chain decompositions.

Connections between geometric configurations and lattice structures are established.

Abstract

Each finite configuration of points in the plane determines a corresponding lattice of noncrossing partitions. When these points form the vertex set of a convex polygon, the associated lattice is the classical noncrossing partition lattice (introduced by Kreweras in 1972), which makes many appearances in combinatorics and geometric group theory. If, on the other hand, all points of the configuration lie on a common line segment, the result is a Boolean lattice. In this article, we examine the more general class of hull configurations, which we define to be those which lie either on a line segment or on the boundary of a convex polygon. We prove that the corresponding lattices of noncrossing partitions are unions of maximal Boolean subposets and, under certain circumstances, have symmetric chain decompositions.