Arrow pattern avoidance in permutations: structure and enumeration

TL;DR

This paper systematically studies arrow pattern avoidance in permutations, establishing structural properties, equivalences, and enumeration results, and explores avoidance of pattern pairs related to fixed points.

Contribution

It introduces arrow-Wilf equivalence, provides enumeration formulas for several classes, and analyzes pairs of arrow patterns, advancing understanding of arrow pattern structures.

Findings

Defined arrow-Wilf equivalence.

Enumerated several arrow avoidance classes.

Analyzed pairs of arrow patterns related to fixed points.

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.