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A Galton-Watson tree approach to local limits of permutations avoiding a pattern of length three

Jungeun Park, Douglas Rizzolo

TL;DR

The paper develops a unified tree-based approach to local limits of permutations avoiding patterns of length three by encoding pattern-avoiding permutations as functions derived from rooted plane trees and analyzing their limits via size-biased Galton–Watson trees. It establishes the convergence of conditioned GW trees to a size-biased limit, constructs explicit bijections between finite/infinite trees and pattern-avoiding permutations for all $oldsymbol{\sigma} obreak\in S_3$, and proves continuity of these bijections under local tree convergence. Consequently, local limits of $oldsymbol{ ext{Av}}_n(oldsymbol{\sigma})$ exist for every length-3 pattern and are described by deterministic functionals on the size-biased GW limit, with a complete description in the case $oldsymbol{\sigma}={f 321}$. This provides a unified framework that connects local permutation limits to classical regenerative and size-biased tree limits, advancing understanding of the limiting objects in pattern-avoiding permutation theory.

Abstract

We use local limits of Galton-Watson trees to establish local limit theorems for permutations conditioned to avoid a pattern of length three. In the case of 321-avoiding permutations our results resolve an open problem of Pinsky. In the other cases our results give new descriptions of the limiting objects in terms of size-biased Galton-Watson trees.

A Galton-Watson tree approach to local limits of permutations avoiding a pattern of length three

TL;DR

The paper develops a unified tree-based approach to local limits of permutations avoiding patterns of length three by encoding pattern-avoiding permutations as functions derived from rooted plane trees and analyzing their limits via size-biased Galton–Watson trees. It establishes the convergence of conditioned GW trees to a size-biased limit, constructs explicit bijections between finite/infinite trees and pattern-avoiding permutations for all , and proves continuity of these bijections under local tree convergence. Consequently, local limits of exist for every length-3 pattern and are described by deterministic functionals on the size-biased GW limit, with a complete description in the case . This provides a unified framework that connects local permutation limits to classical regenerative and size-biased tree limits, advancing understanding of the limiting objects in pattern-avoiding permutation theory.

Abstract

We use local limits of Galton-Watson trees to establish local limit theorems for permutations conditioned to avoid a pattern of length three. In the case of 321-avoiding permutations our results resolve an open problem of Pinsky. In the other cases our results give new descriptions of the limiting objects in terms of size-biased Galton-Watson trees.
Paper Structure (9 sections, 11 theorems, 52 equations)

This paper contains 9 sections, 11 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Tree formalism
  3. Local Limits of Trees
  4. Bijections between rooted plane trees and pattern avoiding permutations
  5. 321/123-avoiding Permutations
  6. 231/213/312/132-avoiding Permutations
  7. Extension to Infinite Trees
  8. Continuity of the bijections
  9. Local limits of pattern avoiding permutations

Key Result

Theorem 1

Fix $\sigma\in S_3$ and suppose that $\Pi^\sigma_n$ is a uniformly random element of $\textbf{A\!v}_n(\sigma)$. Then there exists an $F(\mathbb{N} ,\mathbb{N} ^*)$-valued random variable $\Pi^\sigma$ such that $\Pi^\sigma_n \overset{d}{\longrightarrow} \Pi^\sigma$ as $n\to \infty$.

Theorems & Definitions (19)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Proposition 5
  • proof
  • Definition 6
  • Lemma 3
  • ...and 9 more