The Infinite Limit of Separable Permutations

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 $σ=\{σ_i\}_{ i=1}^\infty$ from $\mathbb{N}$ to $\mathbb{N}^*$ which are injections when restricted to $σ^{-1}(\mathbb{N})$\rm; that is, if $σ_i=σ_j$, $i\neq j$, then $σ_i=\infty$. Extending permutations $σ\in S_n$ by defining $σ_j=j$, 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.

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