Pattern-avoiding permutations and Brownian excursion, Part II: Fixed points

TL;DR

This paper investigates the distribution of fixed points in 123- and 231-avoiding permutations, revealing a connection to Brownian excursion and extending previous research in the field.

Contribution

It provides an exact description of the scaling limit of fixed points in pattern-avoiding permutations using Brownian excursion, building on prior work and Part I of this series.

Findings

Scaling limit of fixed points described by Brownian excursion

Strengthens connections between pattern-avoiding permutations and Brownian motion

Extends recent results on fixed points in pattern-avoiding permutations

Abstract

Permutations that avoid given patterns are among the most classical objects in combinatorics and have strong connections to many fields of mathematics, computer science and biology. In this paper we study fixed points of both 123- and 231-avoiding permutations. We find an exact description for a scaling limit of the empirical distribution of fixed points in term of Brownian excursion. This builds on the connections between pattern-avoiding permutations and Brownian excursion developed in Part I of this series and strengthens the recent results of Elizalde (2012) and Miner and Pak (2014) on fixed points of pattern-avoiding permutations.