The problem of the pawns

S. Kitaev; T. Mansour

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arXiv · math/0305253

The problem of the pawns

S. Kitaev·T. Mansour

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Abstract

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 $\lim_{m,n\to\infty}(M_{m,n})^{1/mn}$ exists and is bounded from above by $(1+\sqrt{5})/2$. 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 $m$.

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S. Kitaev, T. Mansour. 2003-05-18. The problem of the pawns. https://arxiv.org/abs/math/0305253

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