On pattern-avoiding permutons

TL;DR

This paper studies the structure of permutons that avoid a specific permutation, showing they have a simple atomic fiber structure and providing a new proof of the permutation removal lemma.

Contribution

It characterizes the structure of pattern-avoiding permutons and offers a simplified proof of the permutation removal lemma.

Findings

Permutons avoiding a permutation of order k have at most (k-1) atoms in almost every fiber.

The bound of (k-1) atoms is sharp.

A simple proof of the permutation removal lemma is provided.

Abstract

The theory of limits of permutations leads to limit objects called permutons, which are certain Borel measures on the unit square. We prove that permutons avoiding a given permutation of order $k$ have a particularly simple structure. Namely, almost every fiber of the disintegration of the permuton (say, along the x-axis) consists only of atoms, at most $(k-1)$ many, and this bound is sharp. We use this to give a simple proof of the `permutation removal lemma'.