论文标题
圆柱体上的国王置换的poset
The poset of king permutations on a cylinder
论文作者
论文摘要
如果$ | |σ_{i+1} - {i+1} - {i+1} -tentrical king置换}，则置换$σ= [σ_1，\ dots，σ_n] \被称为a {\ em cylindrical king置换}，如果$ |σ_{i+1} -c {i} -s {i} |> 1 $对于每个$ 1 \ leq I \ leq i \ leq i \ leq leq n-1 $ and $ |σ_1-= = 1 $ $> 1 $> 1 $> 1 $> 1 $> 1 $。这个名字来自人们可以看到这些排列的方式，描述了$ n $ kings的位置在$ n \ times n $的棋盘上，以这样的方式（每行和每一列完全包含一个国王和一个国王），没有两个国王互相攻击，额外的条件使国王可以在该行开始时移开一排，并重新启动。 在最近的一篇论文中，我们处理了更一般的“国王排列”集，即仅满足上述两个条件中的第一个。该集合构成了库存众所周知的遏制关系下的诗意。 在本文中，我们研究了圆柱王排列及其结构的子库。我们检查了那些在上层中尽可能大的圆柱王排列。我们使用曼哈顿的距离距离的距离进行了修改及其在圆柱上环境中的某些应用程序，以找到这样的排列标准为$ k-$ ruiltific。我们的主要结果之一是，圆柱排列的两个排列之间的最大差距为$ 4 $。
A permutation $σ=[σ_1,\dots,σ_n] \in S_n$ is called a {\em cylindrical king permutation} if $ |σ_{i+1}-σ_{i}|>1$ for each $1\leq i \leq n-1$ and $|σ_1-σ_n|>1$. The name comes from the the way one can see these permutations as describing locations of $n$ kings on a chessboard of order $n\times n$ in such a way that (each row and each column contains exactly one king and) no two kings are attacking each other, with the additional condition that a king can move off a certain row and reappear at the beginning of that row. In a recent paper, we dealt with the more general set of 'king permutations' i.e. the ones which satisfy only the first of the two conditions above. This set constitutes a poest under the well known containment relation on permutations. In this article we investigate the sub-poset of the cylindrical king permutations and its structure. We examine those cylindrical king permutations whose downset is as large as possible in the upper ranks. We use a modification of Manhattan distance of the plot of a permutation and some of its applications to the cylindrical context to find a criterion for such a permutation to be $k-$ prolific. One of our main results is that the maximal gap between two permutations in the poset of cylindrical permutations is $4$.
