A Note on Noncrossing Path Partitions in the Plane

In [A. García, M. Noy, and J. Tejel, Comput. Geom., 16 (2000), pp. 211–221], it is shown that a double chain of [Formula: see text] points in the plane admits at least [Formula: see text] polygonizations, and it is claimed that it admits at most [Formula: see text] polygonizations. In this note, we provide a proof of this last result. The proof is based on counting noncrossing path partitions for points in the plane in convex position, where a noncrossing 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 [Formula: see text] points in the plane in convex position admits [Formula: see text] noncrossing path partitions and a double chain of [Formula: see text] points in the plane admits at least [Formula: see text] noncrossing path partitions. If isolated points are not allowed, we also show that for [Formula: see text] points in the plane in convex position, there are [Formula: see text] noncrossing path partitions, and for [Formula: see text] points on a double chain, there are at least [Formula: see text] noncrossing path partitions. In addition, using a particular family of noncrossing path partitions for points in convex position, we provide an alternative proof for the result that a double chain of [Formula: see text] points admits at least [Formula: see text] polygonizations.