The infinite limit of separable permutations. Issue 4 (20th May 2021) - Pinsky, Ross G.
Abstract: Let P n sep denote the uniform probability measure on the set of separable permutations in S n . Let ℕ ∗ = ℕ ∪ { ∞ } with an appropriate metric and denote by S ( ℕ, ℕ ∗ ) the compact metric space consisting of functions σ = { σ i } i = 1 ∞ from ℕ to ℕ ∗ which are injections when restricted to σ − 1 ( ℕ ) ; that is, if σ i = σ j, i  ≠  j, then σ i = ∞ . Extending permutations σ ∈ S n by defining σ j = j, for j  >  n, we have S n ⊂ S ( ℕ, ℕ ∗ ) . We show that { P n sep } n = 1 ∞ converges weakly on S ( ℕ, ℕ ∗ ) to a limiting distribution of regenerative type, which we calculate explicitly.