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PreprintCC BY 4.0arXiv

Equidistribution of mesh patterns of length 2 on involutions

  • Xinyu Su
  • Philip B. Zhang
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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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Cite

BibTeXRIS
BibTeX
@misc{su2026equidistribution,
  title = {{Equidistribution of mesh patterns of length 2 on involutions}},
  author = {Xinyu Su and Philip B. Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33028},
  eprint = {2610.33028},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33028},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Equidistribution of mesh patterns of length 2 on involutions
AU  - Xinyu Su
AU  - Philip B. Zhang
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.33028
UR  - https://arxiv.org/abs/2610.33028
AB  - 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.
ER  -