Noncrossing Partitions From Cones and Semicircles

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.