Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Su, Xinyu, Zhang, Philip B.
Institution: Tianjin Normal University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G6545595343
OpenAlex ID: W7223830758