Pattern avoidance in compositions and powers of permutations

TL;DR

This paper introduces a new pattern avoidance concept for compositions and applies it to enumerate permutations avoiding specific pattern chains involving powers, expanding understanding of permutation pattern avoidance.

Contribution

It defines pattern avoidance for compositions and provides enumeration formulas for permutations avoiding certain pattern chains involving powers.

Findings

Enumerates permutations avoiding the chain (312,321:σ) for any pattern σ.

Enumerates permutations avoiding the chain (312,4321:σ) for any σ in S_3.

Extends pattern avoidance concepts to compositions and permutation powers.

Abstract

A permutation $\pi$ is said to avoid a chain $(\sigma:\tau)$ of patterns if $\pi$ avoids $\sigma$ and $\pi^2$ avoids $\tau.$ In this paper, we define a notion of pattern avoidance for compositions of positive integers and use that idea to enumerate permutations of length $n$ that avoid the chain $(312,321:\sigma)$ for any pattern $\sigma\in \bigcup_{m\geq 1} S_m$. We also enumerate those permutations that avoid the chain $(312,4321:\sigma)$ for any $\sigma\in S_3.$