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The Forward Implication

Abstract

Skew-merged permutations have one or two complete rook placements.

Theorem 1.1 (The count for a skew-merged permutation).

Lean statement: D5/S3/Combinatorics/CrosswordGrid/SkewMergedRookForward.forward_count

Proof. Machine-checked in Lean as D5/S3/Combinatorics/CrosswordGrid/SkewMergedRookForward.forward_count (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Joel Brewster Lewis, Robert Won (2026). Non-attacking rook placements on crossword grids. DOI: 10.48550/arXiv.2609.03081. URL: https://arxiv.org/abs/2609.03081v1.

Commentary.

For every positive integer n, if a permutation w of zero through n minus one splits into an increasing and a decreasing subsequence, its permutation grid has exactly one or exactly two complete rook placements.

References