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 $(12;3\to 3)$ and $(21;3\to 3)$ left open by Archer and Laudone, and enumerate the arrow patterns of the form $(ν; b\to c)$ of size $3$ with $ν\in {31, 23, 32}$ and $b,c\in [3]$, 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 $(32;1\to 3)$ unresolved for $|ν|\le 2$, which we pose as an open problem.