  Using T(m,n;k) to denote the number of ways to make a selection of k squares
from an (m x n) rectangular grid with no two squares in the selection adjacent,
we give a formula for T(2,n;k), prove some identities satisfied by these
numbers, and show that T(2,n;k) is given by a degree k polynomial in n. We give
simple formulas for the first few (most significant) coefficients of the
polynomials. We give corresponding results for T(3,n;k) as well. Finally we
prove a unimodality theorem which shows, in particular, how to choose k in
order to maximize T(2,n;k).
