Avoiding patterns with three distinct letters in Canon permutations

@misc{shankar2026avoiding,
  title = {{Avoiding patterns with three distinct letters in Canon permutations}},
  author = {Umesh Shankar},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.30002},
  eprint = {2608.30002},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30002},
}

TY  - UNPB
TI  - Avoiding patterns with three distinct letters in Canon permutations
AU  - Umesh Shankar
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2608.30002
UR  - https://arxiv.org/abs/2608.30002
AB  - We study avoidance of patterns of length $3$ with three distinct letters in canon permutations. We reduce the problem to studying pattern avoidance in lattice words and show that there are $6$ such pattern avoiding classes. This shows that there are $12$ classes for the original Canon permutation pattern avoidance problem. We also give descent refinements for these classes and classify the patterns for which the descent enumeration gives palindromic and γ-positive polynomials. When the polynomials are γ-positive, we explain the γ-positivity through a group action analogous to Foata-Strehl valley hopping. Additionally, we study the avoidance of patterns in the relabelling orbit of $1213, 12112, 1231$ after a conjecture about their cardinalities by Laudone and give bijective proofs for the results.
ER  -