Configuration spaces of hard squares in a rectangle

We study the configuration spaces C(n;p,q) of n labeled hard squares in a p by q rectangle, a generalization of the well-known 15 Puzzle. Our main interest is in the topology of these spaces. Our first result is to describe a cubical cell complex and prove that is homotopy equivalent to the configuration space. We then focus on determining for which n, j, p, and q the homology group $H_j [ C(n;p,q) ]$ is nontrivial. We prove three homology-vanishing theorems, based on discrete Morse theory on the cell complex. Then we describe several explicit families of nontrivial cycles, and a method for interpolating between parameters to fill in most of the picture for large-scale nontrivial homology.