Almost square permutations are typically square

TL;DR

This paper studies almost square permutations with a fixed number of internal points, providing asymptotic enumeration and limit shape results, and extends techniques to 321-avoiding permutations with internal points, linking to Brownian excursions.

Contribution

It introduces a probabilistic approach to asymptotically enumerate almost square permutations and analyzes their limit permutons, also applying methods to 321-avoiding permutations with internal points.

Findings

Asymptotic enumeration of almost square permutations with fixed internal points.

Description of permuton limits for almost square permutations.

Asymptotic enumeration of 321-avoiding permutations with internal points linked to Brownian excursion area.

Abstract

A record in a permutation is a maximum or a minimum, from the left or from the right. The entries of a permutation can be partitioned into two types: the ones that are records are called external points, the others are called internal points. Permutations without internal points have been studied under the name of square permutations. Here, we explore permutations with a fixed number of internals points, called almost square permutations. Unlike with square permutations, a precise enumeration for the total number of almost square permutations of size $n+k$ with exactly $k$ internal points is not known. However, using a probabilistic approach, we are able to determine the asymptotic enumeration. This allows us to describe the permuton limit of almost square permutations with $k$ internal points, both when $k$ is fixed and when $k$ tends to infinity along a negligible sequence with respect to the size of the permutation. Finally, we show that our techniques are quite general by studying the set of $321$-avoiding permutations of size $n$ with exactly $k$ additional internal points ($k$ fixed). In this case we obtain an interesting asymptotic enumeration in terms of the Brownian excursion area. As a consequence, we show that the points of a uniform permutation in this set concentrate on the diagonal and the fluctuations of these points converge in distribution to a biased Brownian excursion.