Fixed points of 321-avoiding permutations

TL;DR

This paper investigates the distribution of fixed points in 321-avoiding permutations by establishing a bijection with rooted plane trees and analyzing their local limits, providing explicit asymptotic descriptions.

Contribution

It introduces a novel bijection between 321-avoiding permutations and rooted plane trees to analyze fixed points distribution.

Findings

Explicit distribution of fixed points in 321-avoiding permutations

Asymptotic limit described via Galton-Watson tree analysis

Provides a new combinatorial approach to pattern-avoiding permutations

Abstract

We describe the distribution of the number and location of the fixed points of permu- tations that avoid the pattern 321 via a bijection with rooted plane trees on n + 1 vertices. Using the local limit theorem for Galton-Watson trees, we are able to give an explicit description of the limit of this distribution.