The problem of the pawns

In this paper we study the number $M_{m,n}$ of ways to place nonattacking pawns on an $m\times n$ chessboard. We find an upper bound for $M_{m,n}$ and analyse its asymptotic behavior. It turns out that ${\lim }_{m,n\to \mathrm{\infty }}(M_{m,n}{)}^{1\mathrm{/}mn}$ exists and is bounded from above by $(1+\sqrt{5})\mathrm{/}2$. Also, we consider a lower bound for $M_{m,n}$ by reducing this problem to that of tiling an $(m+1)\times (n+1)$ board with square tiles of size $1\times 1$ and $2\times 2$. Moreover, we use the transfer-matrix method to implement an algorithm that allows us to get an explicit formula for $M_{m,n}$ for given $m$.