arXiv:2610.33028v1 Announce Type: new Abstract: We classify all 1024 length-two mesh patterns on involutions into exactly 287 equidis- tribution classes, consisting of 262 trivial classes and 25 nontrivial classes. Since many bi- jections on arbitrary permutations do not preserve the involution condition, our proofs use several involution-preserving bijections, including conjugation and local adjacent-transposition operations, transposition of the RSK recording tableau, prefix-rank swap bijections and marked-occurrence bijections. In particular, we establish the equidistribution of INV-099, a nontrivial class closely related to the unrestricted Class 69, by adapting the marked- occurrence bijection of Zhang and Zhao through a symmetric closure of the marked set. We also obtain explicit distribution formulas and recurrences for several classes, including continued fractions obtained by specializing known labeled Motzkin-path formulas and rational expressions derived from paired skew decompositions of involutions.
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