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arXiv · 2603.04218

Arrow pattern avoidance in permutations: structure and enumeration

Kassie Archer·Robert P. Laudone
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Abstract

Arrow patterns were introduced by Berman and Tenner as a generalization of vincular patterns. They observed that arrow patterns have the potential to bridge the divide between a permutation's cycle notation and its one-line notation; in support of this, they used arrow avoidance to enumerate shallow and cyclic shallow permutations. More recently, $321$-avoiding cyclic permutations were recharacterized entirely in terms of arrow avoidance. Motivated by these results, we initiate a systematic study of arrow avoidance. In this paper, we prove structural results about arrow patterns, including defining arrow-Wilf equivalence, and enumerate several arrow avoidance classes. Finally, we consider the avoidance of pairs of arrow patterns, focusing on cases that prohibit fixed points in the underlying permutation.

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Kassie Archer, Robert P. Laudone. 2026-03-04. Arrow pattern avoidance in permutations: structure and enumeration. https://arxiv.org/abs/2603.04218

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