Characterizing avoidance in cycles via vincular patterns

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.