Joint Symmetry of the Mesh Pattern Pair S21

Abstract

The mesh patterns 123 and 321 with the common shading R = {0,1,2} squared together with {(3,3)} have a symmetric joint occurrence distribution.

Theorem 1.1 (Joint equidistribution).

Lean statement: D5/S3/Combinatorics/MeshPattern/MeshPatternS21.result

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MeshPattern/MeshPatternS21.result (✓ std3). ∎

Resolves. Problems/lv-zhang-mesh-123-321-joint-symmetry (proved) by D5/S3/Combinatorics/MeshPattern/MeshPatternS21.result.

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.

For all nonnegative integers n, k and l, the number of permutations of one through n having k occurrences of the mesh pattern 123 and l occurrences of the mesh pattern 321 equals the number having l occurrences of 123 and k occurrences of 321, where both patterns have shading R = {0,1,2} squared together with {(3,3)}.

References

• Truth anchor: D5/S3/Combinatorics/MeshPattern/MeshPatternS21.result
• Dependency: D5/S3/Combinatorics/MeshPattern/MeshPatternS21Switch