Noncrossing Partitions From Cones and Semicircles

TL;DR

This paper explores variations of noncrossing partition lattices derived from point configurations in the plane, extending classical structures to include points on polygon sides.

Contribution

It introduces three new lattice variations based on configurations with points on polygon sides, expanding classical noncrossing partition theory.

Findings

Classical noncrossing partition lattice enumerated by Catalan numbers

New lattice variations accommodate points on polygon sides

Extended properties of these generalized lattices are analyzed

Abstract

For each finite configuration of distinct points in the plane, there is an associated lattice of noncrossing partitions. When these points form the vertices of a convex polygon, the result is the classical noncrossing partition lattice, which is enumerated by the Catalan numbers and satisfies many other useful properties. In this article, we examine three variations of this lattice which arise when the starting configuration is allowed to have points on the sides of a convex polygon rather than just the vertex set.