一个$q$ -皇后问题。IV。攻击型分母
A $q$-Queens Problem. IV. Attacking Configurations and Their Denominators
摘要：
在第一至第三部分中，我们证明了在$n\times n$棋盘上放置$q$非攻击皇后或类似棋子的方法的数目是$n$的拟多项式函数，其系数本质上是$q$中的多项式。在这一部分中，我们集中在这些拟多项式的周期。我们计算由内而外的多面体的顶点的周期因子，因为周期是由它们的最小公分母限定的，并且在理论上等于它们的最小公分母。我们找到一个精确的公式，分母的每一个棋子与一个移动和两个移动棋子有一个水平移动。对于有三个或更多移动的棋子，我们产生与斐波那契数相关的几何结构，这些几何结构显示分母至少随$q$呈指数增长。
英文摘要：
In Parts I-III we showed that the number of ways to place $q$ nonattacking queens or similar chess pieces on an $n\times n$ chessboard is a quasipolynomial function of $n$ whose coefficients are essentially polynomials in $q$. In this part we focus on the periods of those quasipolynomials. We calculate denominators of vertices of the inside-out polytope, since the period is bounded by, and conjecturally equal to, their least common denominator. We find an exact formula for that denominator of every piece with one move and of two-move pieces having a horizontal move. For pieces with three or more moves, we produce geometrical constructions related to the Fibonacci numbers that show the denominator grows at least exponentially with $q$.
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