Pattern avoidance in compositions and powers of permutations

TL;DR

The paper investigates chain-avoidance in permutations by reducing the problem to composition-avoidance, showing that counting permutations $\pi$ with $\pi$ avoiding $312$ and $\pi^2$ avoiding a pattern $\sigma$ can be translated into counting compositions of $n$ that avoid a derived set $C(\sigma)$. For the chain $(312,321: \sigma)$ with $\sigma$ of any length, the authors classify all relevant $\sigma$ via $\Omega$ and provide exact counts in terms of Fibonacci, Tribonacci, and related $k$-nacci sequences, including generating functions and zeroes beyond certain sizes. They also treat the chain $(312,4321: \sigma)$ for $\sigma \in \mathcal{S}_3$, obtaining explicit recurrences and closed forms (or generating functions) for each case, often via a direct-sum decomposition of the permutation and analysis of $\pi^2$. The work thus connects chain-avoidance in permutations to a robust framework of composition-avoidance and yields several complete enumerations, plus several conjectures and directions for higher powers and longer chains with potential applications to pattern-avoidance theory and combinatorial enumeration. The methods provide precise, computable counts and generating functions that reveal deep connections between pattern-avoidance and generalized Fibonacci-type sequences.

Abstract

A permutation $π$ is said to avoid a chain $(σ:τ)$ of patterns if $π$ avoids $σ$ and $π^2$ avoids $τ.$ 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 $(312,321:σ)$ for any pattern $σ\in \bigcup_{m\geq 1} \mathcal{S}_m$. We also enumerate those permutations that avoid the chain $(312,4321:σ)$ for any $σ\in\mathcal{S}_3.$

Theorems & Definitions (32)

• Theorem 2.1
• proof
• Example 2.2
• Example 2.3
• Theorem 2.4
• proof
• Example 2.5
• Lemma 3.1
• proof
• Lemma 3.2