Permuton limits for some permutations avoiding a single pattern

TL;DR

This paper investigates the limiting behavior of permutations avoiding a specific pattern, showing that some converge to a permuton supported on the anti-diagonal, and develops general properties to facilitate such analyses.

Contribution

It introduces new methods for analyzing permuton limits of pattern-avoiding permutations, especially those converging to the anti-diagonal, and establishes general properties of permutons.

Findings

Permuton limits can collapse to the anti-diagonal line in the unit square.

General properties of permutons are identified to assist in proving such limits.

Some permutation classes have permuton limits supported on the line x + y = 1.

Abstract

Permutons are probability measures on the unit square with uniform marginals that provide a natural way to describe limits of permutations. We are interested in the permuton limits for permutations sampled uniformly from certain pattern-avoiding classes that are in bijection with the class of permutations avoiding the increasing pattern of length $d+1$. In particular, we will look at a family of permutations whose permuton limit collapses to the unique permuton supported on the line $x + y = 1$ in the unit square, informally known as the anti-diagonal. We prove some general properties about permutons to aid our efforts, which may be useful for proving permuton limits that converge to the anti-diagonal for a broader range of permutation classes.