发表机构
The George Washington University（乔治华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文定义了纵横字谜网格上的非攻击车放置，证明了一般网格车放置数的上界，揭示稀疏网格车放置与交替符号矩阵的双射关系，还通过Robinson–Schensted对应刻画排列网格的车放置情况，并提出多个猜想与开放问题。
AI 中文摘要
我们引入了纵横字谜网格上非攻击车放置的概念。纵横字谜网格由白格（构成横向和纵向单词）与黑格（分隔单词）组成，该网格上的完全非攻击车放置是白格的一个子集，它与每一个横向单词和每一个纵向单词恰好各相交一次。我们证明了一般网格可容纳的车放置数量的上界。接着我们研究无两个黑格共边的稀疏网格，证明某些稀疏网格上的车放置与带有指定-1项的交替符号矩阵存在双射对应。进一步针对排列网格，我们证明每个排列网格至少容纳一个车放置，并通过Robinson–Schensted对应刻画了其网格恰好容纳一个放置的排列。全文提出了若干猜想与开放问题。
英文摘要
We introduce the notion of a non-attacking rook placement on a crossword grid. A crossword grid is a collection of white squares (which comprise across and down words) and black squares (which separate the words), and a complete non-attacking rook placement on such a grid is a subset of white squares which intersects every across and every down word exactly once. We prove an upper bound on the number of rook placements that a general grid can admit. We then study sparse grids in which no two black squares share an edge and show that rook placements on certain sparse grids correspond bijectively to alternating sign matrices with prescribed $-1$ entries. Specializing further to permutation grids, we prove that every permutation grid admits at least one rook placement, and characterize the permutations whose grids admit exactly one placement in terms of the Robinson--Schensted correspondence. Throughout, we pose a variety of conjectures and open questions.
