Enumerating Pattern-Avoiding Involutions using Combinatorial Exploration

@misc{bean2026enumerating,
  title = {{Enumerating Pattern-Avoiding Involutions using Combinatorial Exploration}},
  author = {Christian Bean and Anthony J. Guttmann and Jay Pantone},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04352},
  eprint = {2609.04352},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04352},
}

TY  - UNPB
TI  - Enumerating Pattern-Avoiding Involutions using Combinatorial Exploration
AU  - Christian Bean
AU  - Anthony J. Guttmann
AU  - Jay Pantone
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.04352
UR  - https://arxiv.org/abs/2609.04352
AB  - The enumeration of pattern-avoiding permutations has been a popular area of study over the past several decades, but comparatively little attention has been given to the topic of pattern-avoiding involutions. In this paper, we derive the algebraic generating functions of two Wilf-equivalence classes of involutions avoiding a single pattern of length $4$, Av^I(2431) and Av^I(3421). We then adapt the Mosaic method, a fast counting algorithm for permutations, to count involutions and apply it to substantially extend the known initial terms of the counting sequences for the remaining two Wilf-equivalence classes avoiding a pattern of length $4$, Av^I(1324) and Av^I(4231). Based on these extended sequences, we empirically analyze the asymptotic behavior of the counting sequences of these two classes.
ER  -