bibkey: fangfukitaevlisusun2026mesh authors: Q. Fang, S. Fu, S. Kitaev, H. Li, X. Su, Z. Sun year: 2026 title: “On mesh patterns of short length: Equidistribution and enumeration” doi: 10.48550/arXiv.2606.14367 url: https://arxiv.org/abs/2606.14367v1 claim: “Concluding remarks, Conjecture 1: the four Class 69 length-2 mesh patterns are equidistributed on involutions.” strata_touched:

• D5/S3/Combinatorics/MeshPattern/Class69InvolutionRefutation license: citation-only triage: anchor

Class 69 mesh patterns on involutions

Verified locator

DOI: 10.48550/arXiv.2606.14367

URL: https://arxiv.org/abs/2606.14367v1

Locator: Concluding remarks, Conjecture 1; the Class 69 pattern list in Remark class-69-remark-equivalence; the preamble definition of the \pattern macro.

Fang, Fu, Kitaev, Li, Su and Sun state in Concluding remarks that the remaining Class 69 equidistribution should continue when restricted to involutions, and Conjecture 1 names the four length-2 mesh patterns \pattern{scale=0.5}{2}{1/1,2/2}{1/2,1/1,2/1,0/0}, \pattern{scale=0.5}{2}{1/1,2/2}{2/2,0/1,1/1,1/0}, \pattern{scale=0.5}{2}{1/1,2/2}{0/2,1/1,2/1,1/0}, and \pattern{scale=0.5}{2}{1/1,2/2}{1/2,0/1,1/1,2/0}. Their macro shades the box whose lower-left corner is each x/y in the fourth argument and places dots at (1,1) and (2,2).

The kernel-checked settlement evaluates the quantified claim at n = 3, k = 0, and patterns R 0 and R 2. Among the involutions 123, 132, 213, and 321, two avoid R 0 while one avoids R 2, so the universal equidistribution claim is false.