We study a family of sorting match puzzles on grids, which we call permutation match puzzles. In this puzzle, each row and column of a $n \times n$ grid is labeled with an ordering constraint -- ascending (A) or descending (D) -- and the goal is to fill the grid with the numbers 1 through $n^2$ such that each row and column respects its constraint. We provide a complete characterization of solvable puzzles: a puzzle admits a solution if and only if its associated constraint graph is acyclic, which translates to a simple "at most one switch" condition on the A/D labels. When solutions exist, we show that their count is given by a hook length formula. For unsolvable puzzles, we present an $O(n)$ algorithm to compute the minimum number of label flips required to reach a solvable configuration. Finally, we consider a generalization where rows and columns may specify arbitrary permutations rather than simple orderings, and establish that finding minimal repairs in this setting is NP-complete by a reduction from feedback arc set.

翻译：我们研究一类基于网格的排序匹配谜题，称之为排列匹配谜题。在此谜题中，$n \times n$ 网格的每一行和每一列均标有排序约束——递增（A）或递减（D）——目标是用数字 1 至 $n^2$ 填充网格，使得每行每列均满足其约束。我们给出了可解谜题的完整刻画：当且仅当其关联约束图是无环的，谜题才存在解，这等价于 A/D 标签满足简单的“至多一次切换”条件。当解存在时，我们证明其数量可由钩长公式给出。对于不可解谜题，我们提出一种 $O(n)$ 算法来计算达到可解配置所需的最小标签翻转次数。最后，我们考虑一种推广形式，其中行和列可指定任意排列而非简单排序，并通过反馈弧集的归约证明在此设定下寻找最小修复方案是 NP 完全的。