Arrow-Wilf equivalences and enumerative results for short arrow patterns

Abstract

Arrow patterns, introduced by Berman and Tenner, provide a unified framework for studying permutation classes where both one-line and cycle structure constraints are present. In this paper, we continue the systematic study of arrow pattern avoidance initiated by Archer and Laudone. We establish several structural results, including a key lemma that translates arrow patterns into vincular patterns under certain conditions, and derive a series of arrow-Wilf equivalences arising from reversal, complementation, and insertion operations. We also resolve the two cases (12;3→ 3) and (21;3→ 3) left open by Archer and Laudone, and enumerate the arrow patterns of the form (ν; b→ c) of size $3$ with ν∈ {31, 23, 32} and b,c∈ [3], providing explicit formulas connecting the results to Bell numbers, Bessel numbers, Catalan numbers, and derangement numbers. Together with earlier work of Archer and Laudone, this leaves only (32;1→ 3) unresolved for |ν|≤ 2, which we pose as an open problem.

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BibTeX RIS

BibTeX

@misc{zhou2026arrowwilf,
  title = {{Arrow-Wilf equivalences and enumerative results for short arrow patterns}},
  author = {Robin D. P. Zhou and Xinyang Yu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29392},
  eprint = {2609.29392},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29392},
}

RIS

TY  - UNPB
TI  - Arrow-Wilf equivalences and enumerative results for short arrow patterns
AU  - Robin D. P. Zhou
AU  - Xinyang Yu
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29392
UR  - https://arxiv.org/abs/2609.29392
AB  - Arrow patterns, introduced by Berman and Tenner, provide a unified framework for studying permutation classes where both one-line and cycle structure constraints are present. In this paper, we continue the systematic study of arrow pattern avoidance initiated by Archer and Laudone. We establish several structural results, including a key lemma that translates arrow patterns into vincular patterns under certain conditions, and derive a series of arrow-Wilf equivalences arising from reversal, complementation, and insertion operations. We also resolve the two cases (12;3→ 3) and (21;3→ 3) left open by Archer and Laudone, and enumerate the arrow patterns of the form (ν; b→ c) of size $3$ with ν∈ {31, 23, 32} and b,c∈ [3], providing explicit formulas connecting the results to Bell numbers, Bessel numbers, Catalan numbers, and derangement numbers. Together with earlier work of Archer and Laudone, this leaves only (32;1→ 3) unresolved for |ν|≤ 2, which we pose as an open problem.
ER  -

Abstract

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