When arrow patterns meet classical patterns
Abstract
Seeking to bridge the structural divide between a permutation's cycle notation and its one-line notation, Berman and Tenner introduced a novel notion of permutation pattern known as the arrow pattern. Recently, Archer and Laudone initiated a systematic study of arrow pattern avoidance, leaving behind three intriguing conjectures. In this paper, we resolve all three conjectures. First, we enumerate all six subclasses of permutations that simultaneously avoid a classical pattern of length 3 and a fixed arrow pattern of length 3, thereby confirming the first two conjectures. Second, we settle the third conjecture (which involves a different arrow pattern) by providing two independent proofs. These proofs rely on a restriction of Biane's bijection to non-nesting involutions and Krattenthaler's bijection from 321-avoiding permutations to Dyck paths, respectively.
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BibTeX
@misc{fu2026when,
title = {{When arrow patterns meet classical patterns}},
author = {Shishuo Fu and Zhenghe Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.04094},
eprint = {2607.04094},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04094},
}
RIS
TY - UNPB
TI - When arrow patterns meet classical patterns
AU - Shishuo Fu
AU - Zhenghe Yang
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2607.04094
UR - https://arxiv.org/abs/2607.04094
AB - Seeking to bridge the structural divide between a permutation's cycle notation and its one-line notation, Berman and Tenner introduced a novel notion of permutation pattern known as the arrow pattern. Recently, Archer and Laudone initiated a systematic study of arrow pattern avoidance, leaving behind three intriguing conjectures. In this paper, we resolve all three conjectures. First, we enumerate all six subclasses of permutations that simultaneously avoid a classical pattern of length 3 and a fixed arrow pattern of length 3, thereby confirming the first two conjectures. Second, we settle the third conjecture (which involves a different arrow pattern) by providing two independent proofs. These proofs rely on a restriction of Biane's bijection to non-nesting involutions and Krattenthaler's bijection from 321-avoiding permutations to Dyck paths, respectively.
ER -