The problem of the kings (Q1346737)

Summary: On a \(2m\times 2n\) chessboard, the maximum number of nonattacking kings that can be placed is \(mn\), since each \(2\times 2\) cell can have at most one king. Let \(f_ m(n)\) denote the number of ways that \(mn\) nonattacking kings can be placed on a \(2m\times 2n\) chessboard. The purpose of this paper is to prove the following result. Theorem. For each \(m= 1,2,3,\dots\) there are constants \(c_ m> 0\), \(d_ m\), and \(0\leq \theta_ m< m+ 1\) such that \(f_ m(n)= (c_ m n+ d_ m)(m+1)^ n+ O(\theta^ n_ m)\) \((n\to \infty)\).