On the number of crossing-free matchings, cycles, and partitions

We show that a set of n points in the plane has at most O(10.05 ⁿ) perfect matchings with crossing-free straight-line embedding. The expected number of perfect crossing-free matchings of a set of n points drawn independently and identically distributed from an arbitrary distribution in the plane is at most O(9.24ⁿ). Several related bounds are derived: (a) The number of all (not necessarily perfect) crossing-free matchings is at most O(10.43ⁿ). (b) The number of red-blue perfect crossing-free matchings (where the points are colored red or blue and each edge of the matching must connect a red point with a blue point) is at most O(7.61ⁿ). (c) The number of left-right perfect crossing-free matchings (where the points are designated as left or right endpoints of the matching edges) is at most O(5.38ⁿ). (d) The number of perfect crossing-free matchings across a line (where all the matching edges must cross a fixed halving line of the set) is at most 4ⁿ. These bounds are employed to infer that a set of n points in the plane has at most O(86.81ⁿ) crossing-free spanning cycles (simple polygonizations) and at most O(12.24ⁿ) crossing-free partitions (these are partitions of the point set so that the convex hulls of the individual parts are pairwise disjoint). We also derive lower bounds for some of these quantities.