ILL 1988 Problem 19

Let $Z_{m,n}$ be the set of all ordered pairs $(i,j)$ with $i \in {1, \ldots, m}$ and $j \in {1, \ldots, n}.$ Also let $a_{m,n}$ be the number of all those subsets of $Z_{m,n}$ that contain no 2 ordered pairs $(i_1,j_1)$ and $(i_2,j_2)$ with $|i_1 - i_2| + |j_1 - j_2| = 1.$ Then show, for all positive integers $m$ and $k,$ that $$a^2_{m, 2 \cdot k} \leq a_{m, 2 \cdot k - 1} \cdot a_{m, 2 \cdot k + 1}.$$