The 15 Puzzle and homological stability in the space direction

The ordered configuration space of $n$ open unit squares in the $w$ by $h$ rectangle exhibits homological stability in the space direction. That is, for fixed $n$ and fixed homological degree $k$, once the underlying rectangle is large enough, making it any larger does not change the $k$-th homology of the square configuration space. In this paper, we sharpen the stable range. Finding bounds for $w$ and $h$ in terms of $n$ and $k$, we prove that most rectangles can be almost entirely filled with squares and there still be an isomorphism between the $k$-th homology of the resulting square configuration space and the $k$-th homology of the ordered configuration space of $n$ points in the plane.