Characterizing avoidance in cycles via vincular patterns

TL;DR

This paper characterizes cyclic permutations avoiding the pattern 321 through vincular pattern avoidance, using pattern functions and arrow patterns, and explores growth rate bounds for such cycles.

Contribution

It introduces a novel characterization of 321-avoiding cycles via vincular patterns and pattern functions, linking permutation avoidance to combinatorial pattern structures.

Findings

Cyclic permutations avoiding 321 are characterized by vincular pattern avoidance.

The paper establishes bounds on the growth rate of 321-avoiding cycles.

It connects pattern avoidance with inequalities in permutation enumeration.

Abstract

We show that cyclic permutations avoiding $321$ are precisely those permutations whose image under the fundamental bijection avoid a set of vincular patterns. We do this by using pattern functions and arrow patterns, in combination with the characterization of $321$ avoidance in terms of equality of the upper bound of the Daiconis-Graham inequalities. We then explore some consequences of this result, including upper and lower bound results on the growth rate of $321$ avoiding cycles.