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

We show that a set of <i>n</i> points in the plane has at most <i>O</i>(10.05<sup><i>n</i></sup>) perfect matchings with crossing-free straight-line embedding. The expected number of perfect crossing-free matchings of a set of <i>n</i> points drawn i.i.d. from an arbitrary distribution in the plane is at most <i>O</i>(9.24<sup><i>n</i></sup>).Several related bounds are derived: (a) The number of all (not necessarily perfect) crossing-free matchings is at most <i>O</i>(10.43<sup><i>n</i></sup>). (b) The number of <i>left-right</i> perfect crossing-free matchings (where the points are designated as left or as right endpoints of the matching edges) is at most <i>O</i>(5.38<sup><i>n</i></sup>). (c) 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 4<sup><i>n</i></sup>.These bounds are employed to infer that a set of <i>n</i> points in the plane has at most <i>O</i>(86.81<sup><i>n</i></sup>) crossing-free spanning cycles (simple polygonizations), and at most <i>O</i>(12.24<sup><i>n</i></sup>) crossing-free partitions (partitions of the point set, so that the convex hulls of the individual parts are pairwise disjoint).

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