The Problem of Kings

Abstract

Let $f(n)$ denote the number of configurations of $n^2$ mutually non-attacking kings on a $2n\times 2n$ chessboard. We show that $\log f(n)$ grows like $2n\log n - 2n\log 2$, with an error term of $O(n^{4/5}\log n)$. The result depends on an estimate for the sum of the entries of a high power of a matrix with positive entries.