On mesh patterns of short length: Equidistribution and enumeration
Abstract
The classification and enumeration of short mesh patterns have emerged as two central directions in the area. We make substantial progress on both fronts. We construct an involution and a bijection that establish distributional equivalences for two classes of length-$2$ mesh patterns, thereby resolving a conjecture from 2019 and a recent conjecture. As a consequence, the best known upper bounds for the numbers of distribution-equivalence and Wilf-equivalence classes drop to $106$ and $47$, respectively. Combined with the known lower bounds of 105 and 46, conjectured to be exact, these results leave both classifications hinging on a single distribution-equivalence question conjectured in 2019, whose resolution would at once settle the remaining Wilf-equivalence case. We further conjecture that this unresolved equidistribution also holds for involutions, a subclass of all permutations. We also determine the distributions of three additional classes of length-$2$ mesh patterns through a detailed structural analysis. Our work combines bijective techniques with generating-function methods, yielding new insights into the structure and enumeration of short mesh patterns.
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BibTeX
@misc{fang2026mesh,
title = {{On mesh patterns of short length: Equidistribution and enumeration}},
author = {Qi Fang and Shishuo Fu and Sergey Kitaev and Haijun Li and Xinyu Su and Ziyao Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.14367},
eprint = {2606.14367},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.14367},
}
RIS
TY - UNPB
TI - On mesh patterns of short length: Equidistribution and enumeration
AU - Qi Fang
AU - Shishuo Fu
AU - Sergey Kitaev
AU - Haijun Li
AU - Xinyu Su
AU - Ziyao Sun
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2606.14367
UR - https://arxiv.org/abs/2606.14367
AB - The classification and enumeration of short mesh patterns have emerged as two central directions in the area. We make substantial progress on both fronts. We construct an involution and a bijection that establish distributional equivalences for two classes of length-$2$ mesh patterns, thereby resolving a conjecture from 2019 and a recent conjecture. As a consequence, the best known upper bounds for the numbers of distribution-equivalence and Wilf-equivalence classes drop to $106$ and $47$, respectively. Combined with the known lower bounds of 105 and 46, conjectured to be exact, these results leave both classifications hinging on a single distribution-equivalence question conjectured in 2019, whose resolution would at once settle the remaining Wilf-equivalence case. We further conjecture that this unresolved equidistribution also holds for involutions, a subclass of all permutations. We also determine the distributions of three additional classes of length-$2$ mesh patterns through a detailed structural analysis. Our work combines bijective techniques with generating-function methods, yielding new insights into the structure and enumeration of short mesh patterns.
ER -