  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.
