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We develop a recursive formula for counting the number of rectangulations ofa square, i.e the number of combinatorially distinct tilings of a square byrectangles. Our formula specializes to give a formula counting genericrectangulations, as analyzed by Reading in [5]. Our computations agree with [5]as far as was calculated and extend to the non-generic case. An interestingfeature of the number of rectangulations is that it appears to have an 8-foldperiodicity modulo 2. We verify this periodicity for small values of n, but thegeneral result remains elusive, perhaps hinting at some unseen structure on thespace of rectangulations, analogous to Reading's discovery that genericrectangulations 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 lessthan or equal to n rectangles is homotopy-equivalent to a wedge of some numberof (n-1)-dimensional spheres. Combined with formulae for the number of tilings,the exact homotopy type is computed for $n\leq 28$.

On the number of tilings of a square by rectangles