Point sets with many non-crossing perfect matchings

The maximum number of non-crossing straight-line perfect matchings that a set of n points in the plane can have is known to be O(10.0438) and Ω∗(3n). The lower bound, due to Garćıa, Noy, and Tejel (2000), is attained by the double chain, which has Θ(3/n) such matchings. We reprove this bound in a simplified way that uses the novel notion of down-free matchings. We then apply this approach to several other constructions. As a result, we improve the lower bound. First we show that the double zigzag chain with n points has Θ∗(λn) non-crossing perfect matchings with λ ≈ 3.0532. Next we analyze further generalizations of double zigzag chains – double r-chains. The best choice of parameters leads to a construction that has Θ∗(νn) matchings with ν ≈ 3.0930. The derivation of this bound requires an analysis of a coupled dynamic-programming recursion between two infinite vectors.