n皇后问题及相关问题的封闭表达式
Closed-Form Expressions for the n-Queens Problem and Related Problems
摘要：
本文利用$(0,1)$矩阵的积和式，导出了$n$ -queens问题及三个相关问题的简单封闭形式.这些公式是同类中的第一个。此外，他们提供了解决这些问题的多项式空间，有一个非平凡的时间复杂度界的第一种方法。然后，我们将展示如何封闭形式的拉丁广场的数量顺序$n$如下从我们的方法。最后，我们证明了下界。特别是，我们表明，永久的舒尔的复值矩阵是一个下界的环形半皇后问题，或等价地，在循环拉丁方的横截数。
英文摘要：
In this paper, we derive simple closed-form expressions for the $n$-queens problem and three related problems in terms of permanents of $(0,1)$ matrices. These formulas are the first of their kind. Moreover, they provide the first method for solving these problems with polynomial space that has a nontrivial time complexity bound. We then show how a closed-form for the number of Latin squares of order $n$ follows from our method. Finally, we prove lower bounds. In particular, we show that the permanent of Schur's complex valued matrix is a lower bound for the toroidal semi-queens problem, or equivalently, the number of transversals in a cyclic Latin square.
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