一个$q$ -皇后问题。III。非攻击型偏皇后
A $q$-Queens Problem. III. Nonattacking Partial Queens
摘要：
我们应用我们的几何理论来计算$q$在$n\times n$棋盘上的非攻击性位置，从第一部分到第二部分，部分皇后：也就是说，棋子的水平，垂直和$45^\circ$对角移动的任何组合。第一部分和第二部分表明，对于任何骑手（一件移动无限长）的答案将是一个准多项式函数的$n$中的系数基本上是多项式在$q$。这些一般性结果给出了计数拟多项式的三个最高阶系数和两个非攻击棋子的位置计数公式以及组合上不同的这种位置类型。相比之下，我们在这里提出的统一框架，部分皇后允许我们显式计算的四个最高阶系数的计数拟多项式，显示，五个最高阶系数是常数（独立于$n$），并找到下一个系数的周期（这取决于确切的一组移动）。此外，对于三个非攻击部分皇后，我们能够证明非攻击位置的总数和它们的组合不同类型的数量的公式。证明的方法，在前面的部分中，是通过详细分析由内而外的多面体的子空间的格。
英文摘要：
We apply our geometrical theory for counting placements of $q$ nonattacking on an $n\times n$ chessboard, from Parts~I and II, to partial queens: that is, chess pieces with any combination of horizontal, vertical, and $45^\circ$-diagonal moves. Parts~I and II showed that for any rider (a piece with moves of unlimited length) the answer will be a quasipolynomial function of $n$ in which the coefficients are essentially polynomials in $q$. Those general results gave the three highest-order coefficients of the counting quasipolynomial and formulas for counting placements of two nonattacking pieces and the combinatorially distinct types of such placements. By contrast, the unified framework we present here for partial queens allows us to explicitly compute the four highest-order coefficients of the counting quasipolynomial, show that the five highest-order coefficients are constant (independent of $n$), and find the period of the next coefficient (which depends upon the exact set of moves). Furthermore, for three nonattacking partial queens we are able to prove formulas for the total number of nonattacking placements and for the number of their combinatorially distinct types. The method of proof, as in the previous parts, is by detailed analysis of the lattice of subspaces of an inside-out polytope.
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