Bimonotone Subdivisions of Point Configurations in the Plane

TL;DR

This paper studies bimonotone subdivisions of grid point configurations, providing formulas, algorithms, and connections to combinatorial numbers, and explores their properties and enumeration in the plane.

Contribution

It introduces recursions, formulas, and algorithms for counting bimonotone subdivisions and triangulations of grid configurations, linking them to Schr"oder numbers and flip connectivity.

Findings

Number of bimonotone subdivisions relates to large Schr"oder numbers.

Algorithms for counting bimonotone subdivisions of any grid size.

All bimonotone triangulations of a grid are connected by flips.

Abstract

Bimonotone subdivisions in two dimensions are subdivisions all of whose sides are either vertical or have nonnegative slope. They correspond to statistical estimates of probability distributions of strongly positively dependent random variables. The number of bimonotone subdivisions compared to the total number of subdivisions of a point configuration provides insight into how often the random variables are positively dependent. We give recursions as well as formulas for the numbers of bimonotone and total subdivisions of $2\times n$ grid configurations in the plane. Furthermore, we connect the former to the large Schr\"oder numbers. We also show that the numbers of bimonotone and total subdivisions of a $2\times n$ grid are asymptotically equal. We then provide algorithms for counting bimonotone subdivisions for any $m \times n$ grid. Finally, we prove that all bimonotone triangulations of an $m \times n$ grid are connected by flips. This gives rise to an algorithm for counting the number of bimonotone (and total) triangulations of an $m\times n$ grid.