A note on non-crossing path partitions in the plane

In the paper ``Lower bounds on the number of crossing-free subgraphs of $K_N$'' (Computational Geometry 16 (2000), 211-221), it is shown that a double chain of $n$ points in the plane admits at least $\Omega(4.642126305^n)$ polygonizations, and it is claimed that it admits at most $O(5.61^n)$ polygonizations. In this note, we provide a proof of this last result. The proof is based on counting non-crossing path partitions for points in the plane in convex position, where a non-crossing path partition consists of a set of paths connecting the points such that no two edges cross and isolated points are allowed. We prove that a set of $n$ points in the plane in convex position admits $\mathcal{O}^*(5.610718614^{n})$ non-crossing path partitions and a double chain of $n$ points in the plane admits at least $\Omega(7.164102920^n)$ non-crossing path partitions. If isolated points are not allowed, we also show that there are $\mathcal{O}^*(4.610718614^n)$ non-crossing path partitions for $n$ points in the plane in convex position and at least $\Omega(6.164492582^n)$ non-crossing path partitions in a double chain of $n$ points in the plane. In addition, using a particular family of non-crossing path partitions for points in convex position, we provide an alternative proof for the result that a double chain of $n$ points admits at least $\Omega(4.642126305^n)$ polygonizations.