The runsort permuton

TL;DR

This paper proves that the scaled plots of permutations obtained by sorting the ascending runs of a random permutation converge to a specific permuton with an explicitly described support, answering a recent open question.

Contribution

It establishes the convergence of the runsort permutation plots to a new, explicitly characterized permuton, advancing understanding of permutation limit shapes.

Findings

The scaled plots converge to a permuton with probability 1.

The support of the limiting permuton is explicitly described as rom the abstract.

The limit shape is characterized by the inequality rom the support description.

Abstract

Suppose we choose a permutation $\pi$ uniformly at random from $S_n$. Let $\mathsf{runsort}(\pi)$ be the permutation obtained by sorting the ascending runs of $\pi$ into lexicographic order. Alexandersson and Nabawanda recently asked if the plot of $\mathsf{runsort}(\pi)$, when scaled to the unit square $[0,1]^2$, converges to a limit shape as $n\to\infty$. We answer their question by showing that the measures corresponding to the scaled plots of these permutations $\mathsf{runsort}(\pi)$ converge with probability $1$ to a permuton (limiting probability distribution) that we describe explicitly. In particular, the support of this permuton is $\{(x,y)\in[0,1]^2:x\leq ye^{1-y}\}$.