The Infinite Limit of Random Permutations Avoiding Patterns of Length Three

TL;DR

This paper investigates the limiting behavior of uniformly random permutations avoiding patterns of length three, extending the analysis to an infinite setting and describing the convergence of associated probability measures.

Contribution

It introduces a framework for studying the infinite limits of pattern-avoiding permutations and characterizes their behavior in a compact metric space.

Findings

Describes the convergence of measures for permutations avoiding each pattern in S_3.

Provides a new perspective on the structure of infinite pattern-avoiding permutations.

Establishes a foundation for analyzing infinite permutation limits in a rigorous metric space.

Abstract

For $\tau\in S_3$, let $\mu_n^{\tau}$ denote the uniformly random probability measure on the set of $\tau$-avoiding permutations in $S_n$. Let $\mathbb{N}^*=\mathbb{N}\cup\{\infty\}$ with an appropriate metric and denote by $S(\mathbb{N},\mathbb{N}^*)$ the compact metric space consisting of functions $\sigma=\{\sigma_i\}_{ i=1}^\infty$ from $\mathbb{N}$ to $\mathbb{N}^*$ which are injections when restricted to $\sigma^{-1}(\mathbb{N})$\rm; that is, if $\sigma_i=\sigma_j$, $i\neq j$, then $\sigma_i=\infty$. Extending permutations $\sigma\in S_n$ by defining $\sigma_j=j$, for $j>n$, we have $S_n\subset S(\mathbb{N},\mathbb{N}^*)$. For each $\tau\in S_3$, we study the limiting behavior of the measures $\{\mu_n^{\tau}\}_{n=1}^\infty$ on $S(\mathbb{N},\mathbb{N}^*)$.