On the number of tilings of a square by rectangles

We develop a recursive formula for counting the number of rectangulations of a square, i.e the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations, as analyzed by Reading in [5]. Our computations agree with [5] as far as was calculated and extend to the non-generic case. An interesting feature of the number of rectangulations is that it appears to have an 8-fold periodicity modulo 2. We verify this periodicity for small values of n, but the general result remains elusive, perhaps hinting at some unseen structure on the space of rectangulations, analogous to Reading's discovery that generic rectangulations are in 1-1 correspondence with a certain class of permutations. Finally, we use discrete Morse theory to show that the space of tilings by less than or equal to n rectangles is homotopy-equivalent to a wedge of some number of (n-1)-dimensional spheres. Combined with formulae for the number of tilings, the exact homotopy type is computed for $n\leq 28$.