Distributions of Mesh Patterns of Short Lengths on Separable Permutations
Abstract
This paper contributes to the long line of research on the distribution of mesh patterns in permutations. We extend these studies to separable permutations and carry out a comprehensive analysis of mesh patterns of length at most~2. For mesh patterns of length~1, we determine the distributions for all six equivalence classes. In addition, we obtain the joint distribution for the patterns in the class containing the well-known permutation statistic known as the strict fixed point. Furthermore, computer experiments suggest that at most 124 pairs of mesh patterns of type 12 and 21 with identical shading are equidistributed. Using symmetry operations, we partition these pairs into 38 equivalence classes with respect to distribution. By explicitly determining 24 of these distributions (in many cases, in fact, joint equidistributions of the respective pairs), we reduce the number of equivalence classes to~31, which turns out to be the true number of equivalence classes. Enumerating 14 of these classes is left as an open problem.
Related
- A p-adic Valuation Calculus for Permutations with Restricted Cycle Lengths
- A Short Proof of Zhang's Conjecture on Stack-Sortable Permutations
- Counting Hamiltonian Sturm permutations: generating functions and Gaussian distributions
- Two refined Wilf-equivalences for patterns of length 4 and a conjecture on Dumont permutations of the first kind
- Avoiding patterns with three distinct letters in Canon permutations
BibTeX
@misc{gao2026distributions,
title = {{Distributions of Mesh Patterns of Short Lengths on Separable Permutations}},
author = {Alice L. L. Gao and Sergey Kitaev and Ya-Xing Li and Yun Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08865},
eprint = {2610.08865},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08865},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Distributions of Mesh Patterns of Short Lengths on Separable Permutations
AU - Alice L. L. Gao
AU - Sergey Kitaev
AU - Ya-Xing Li
AU - Yun Li
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.08865
UR - https://arxiv.org/abs/2610.08865
AB - This paper contributes to the long line of research on the distribution of mesh patterns in permutations. We extend these studies to separable permutations and carry out a comprehensive analysis of mesh patterns of length at most~2. For mesh patterns of length~1, we determine the distributions for all six equivalence classes. In addition, we obtain the joint distribution for the patterns in the class containing the well-known permutation statistic known as the strict fixed point. Furthermore, computer experiments suggest that at most 124 pairs of mesh patterns of type 12 and 21 with identical shading are equidistributed. Using symmetry operations, we partition these pairs into 38 equivalence classes with respect to distribution. By explicitly determining 24 of these distributions (in many cases, in fact, joint equidistributions of the respective pairs), we reduce the number of equivalence classes to~31, which turns out to be the true number of equivalence classes. Enumerating 14 of these classes is left as an open problem.
ER -