Coincidental pattern avoidance

TL;DR

This paper investigates the conditions under which different types of permutation pattern avoidance, specifically barred and vincular patterns, coincide, providing a comprehensive classification of such cases.

Contribution

It determines when barred pattern avoidance coincides with avoiding finite sets of vincular patterns and vice versa, answering a question posed by Steingrimsson.

Findings

720 barred patterns with this property identified

48 vincular patterns with this property identified

Conditions for coincidence between barred and vincular pattern avoidance established

Abstract

There are several versions of permutation pattern avoidance that have arisen in the literature, and some known examples of two different types of pattern avoidance coinciding. In this paper, we examine barred patterns and vincular patterns. Answering a question of Steingrimsson, we determine when barred pattern avoidance coincides with avoiding a finite set of vincular patterns, and when vincular pattern avoidance coincides with avoiding a finite set of barred patterns. There are 720 barred patterns with this property, each having between 3 and 7 letters, of which at most 2 are barred, and there are 48 vincular patterns with this property, each having between 2 and 4 letters and exactly one bond.