bibkey: lvzhang2025mesh authors: Shuzhen Lv, Philip B. Zhang year: 2025 title: “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 claim: “Conjecture 2 (pair S21) on the joint equidistribution of the mesh patterns 123 and 321 with a common symmetric shading.” strata_touched:

• D5/S3/Combinatorics/MeshPattern/MeshPatternS21 license: citation-only triage: anchor

Lv and Zhang, joint equidistributions of mesh patterns 123 and 321

The paper studies pairs of mesh patterns 123 and 321 carrying the same shading and proves joint equidistributions for symmetric and minus-antipodal shadings, leaving two pairs as a conjecture.

Section 2.4, Conjecture 2, printed page 9, states for the pairs S21 and S22 that the mesh patterns 123 and 321 with the displayed shadings are jointly equidistributed. The shadings are given only as diagrams. For S21 the rendered page shows the shaded cells R = {0,1,2}² ∪ {(3,3)}, where a cell (i, j) has column i and row j counted from the lower left. The source proves only the Wilf-equivalence of the four patterns (Theorem 3) and notes that S22 is obtained from S21 by reverse and complement.

Verified locator

DOI: 10.48550/arXiv.2501.00357

URL: https://arxiv.org/abs/2501.00357v3

• Locator: Section 2.4, Conjecture 2, printed page 9 (pair S21 and pair S22, shadings given as diagrams).
• Locator: Section 2.4, Theorem 3, printed page 9 (Wilf-equivalence of the four patterns).
• Locator: Section 1, cell convention for mesh patterns.