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 $\sigma=\{\sigma_i\}_{ i=1}^\infty$ 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.