The Infinite Limit of Separable Permutations

TL;DR

This paper studies the limiting behavior of uniform measures on separable permutations as their size grows, showing convergence to a explicitly characterized regenerative distribution in an extended permutation space.

Contribution

It introduces a new framework for analyzing the asymptotic distribution of separable permutations and explicitly computes the limiting regenerative distribution.

Findings

Weak convergence of measures on an extended permutation space

Explicit formula for the limiting regenerative distribution

Characterization of the infinite limit of separable permutations

Abstract

Let $P_n^{\text{sep}}$ denote the uniform probability measure on the set of separable permutations in $S_n$. Let $\mathbb{N}^*=\mathbb{N}\cup\{\infty\}$ with an appropriate metric and denote by $S(\mathbb{N},\mathbb{N}^*)$ the compact metric space consisting of functions $\sigma=\{\sigma_i\}_{ i=1}^\infty$ from $\mathbb{N}$ to $\mathbb{N}^*$ which are injections when restricted to $\sigma^{-1}(\mathbb{N})$\rm; that is, if $\sigma_i=\sigma_j$, $i\neq j$, then $\sigma_i=\infty$. Extending permutations $\sigma\in S_n$ by defining $\sigma_j=j$, for $j>n$, we have $S_n\subset S(\mathbb{N},\mathbb{N}^*)$. We show that $\{P_n^{\text{sep}}\}_{n=1}^\infty$ converges weakly on $S(\mathbb{N},\mathbb{N}^*)$ to a limiting distribution of regenerative type, which we calculate explicitly.