On the number of rectangulations of a planar point set

We investigate the number of different ways in which a rectangle containing a set of n noncorectilinear points can be partitioned into smaller rectangles by n (nonintersecting) segments, such that every point lies on a segment. We show that when the relative order of the points forms a separable permutation, the number of rectangulations is exactly the ( n + 1 ) st Baxter number. We also show that no matter what the order of the points is, the number of guillotine rectangulations is always the nth Schrِder number, and the total number of rectangulations is O ( 20 n / n 4 ) .