Vol. 12, No. 2, 2021

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Bimonotone subdivisions of point configurations in the plane

Elina Robeva and Melinda Sun

Vol. 12 (2021), No. 2, 125–138
DOI: 10.2140/astat.2021.12.125
Abstract

Bimonotone subdivisions in two dimensions are subdivisions all of whose sides are 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 × n grid configurations in the plane. Furthermore, we connect the former to the large Schröder numbers. We also show that the numbers of bimonotone and total subdivisions of a 2 × n grid are asymptotically equal. We then provide algorithms for counting bimonotone subdivisions for any m × n grid. Finally, we prove that all bimonotone triangulations of an m × n grid are connected by flips. This gives rise to an algorithm for counting the number of bimonotone (and total) triangulations of an m × n grid.

Keywords
bimonotone, subdivisions, triangulations, log-concave densities, log-supermodular densities, MTP${}_2$
Mathematical Subject Classification
Primary: 52C05
Milestones
Received: 13 September 2020
Revised: 10 August 2021
Accepted: 28 August 2021
Published: 13 December 2021
Authors
Elina Robeva
Department of Mathematics
University of British Columbia
Vancouver, BC
Canada
Melinda Sun
Massachusetts Institute of Technology
Cambridge, MA
United States