Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

Transposition Inside the Active Region

Abstract

Transposing the labels on an active boundary produces a permutation with the same active rectangles and the same points outside the active region.

Theorem 1.1 (The active-region switch).

Lean statement: D5/S3/Combinatorics/MeshPattern/MeshPatternS21Switch.seven_active_switch

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MeshPattern/MeshPatternS21Switch.seven_active_switch (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Shuzhen Lv, Philip B. Zhang (2025). Joint Equidistributions of Mesh Patterns 123 and 321 with Symmetric and Minus-Antipodal Shadings. DOI: 10.48550/arXiv.2501.00357. URL: https://arxiv.org/abs/2501.00357v3.

Commentary.

Let p be a permutation of one through n. Its active outline has n downward steps and a cell list without repetitions covering exactly the active region. Peeling its labeled boundary returns the entries of p on those cells and an axis path with only empty labels. Transposing every boundary label yields a boundary whose peeling returns the entries of another permutation q of one through n and an axis path with only empty labels. The permutations agree outside the active region. For every m from three through n, their rectangles of width n minus m plus three and height m have equal point counts, agree on whether they contain a point of value m, and agree on whether their last column contains a point. They have the same active heights and active region.

References