Cyclic permutations avoiding pairs of patterns of length three

TL;DR

This paper provides explicit formulas for counting cyclic permutations avoiding pairs of length-three patterns, completing the enumeration for most pairs, and establishes a growth rate lower bound for permutations avoiding certain single patterns.

Contribution

It offers new explicit formulas for most pattern pairs and a lower bound for growth rates in cyclic permutations avoiding specific patterns.

Findings

Explicit formulas for most pattern pairs avoiding length-three patterns

The pair (123,231) is identified as the most challenging case

A lower bound for the growth rate of permutations avoiding certain patterns

Abstract

We complete the enumeration of cyclic permutations avoiding two patterns of length three each by providing explicit formulas for all but one of the pairs for which no such formulas were known. The pair $(123,231)$ proves to be the most difficult of these pairs. We also prove a lower bound for the growth rate of the number of cyclic permutations that avoid a single pattern $q$, where $q$ is an element of a certain infinite family of patterns.