Tiling Parity Results and the Holey Square Solution

The number of domino tilings of the 2n× 2n square with a centered hole of size 2m×2m, a figure known as the holey square and denoted H(m, n), was conjectured by Edward Early to be 2n−m(2km,n + 1) . Although the conjecture has remained unsolved until now, specific cases were known, for example see [4]. In this paper, we answer the general conjecture in the affirmative, primarily via a theorem about tiling parity that has applications beyond the problem of the holey square. We also give combinatorial meaning to the odd factor 2km,n + 1 in Early’s conjecture, and demonstrate other consequences of the parity theorem.