On the number of crossing-free partitions

Abstract

A partition of a point set in the plane is called crossing-free, if the convex hulls of the individual parts do not intersect. We prove that convex position of a planar set of n points in general position minimizes the number of crossing-free partitions into 1, 2, 3, and n-3, n-2, n-1, n partition classes. Moreover, we show that for all n≥5 convex position of the underlying point set does not maximize the total number of crossing-free partitions. It is known that in convex position the number of crossing-free partitions into k classes equals the number of partitions into n-k+1 parts. This does not hold in general, and we mention a construction for point sets with significantly more partitions into few classes than into many. © 2011 Elsevier B.V. All rights reserved.