The infinite limit of separable permutations

• Published in: Random structures & algorithms Vol. 59; no. 4; pp. 622 - 639
• Main Author: Pinsky, Ross G.
• Format: Journal Article
• Language: English
• Published: New York John Wiley & Sons, Inc 01-12-2021; Wiley Subscription Services, Inc
• Subjects: Metric space; Permutations; random permutation; Schröder numbers; separable permutations
• ISSN: 1042-9832; 1098-2418
• DOI: 10.1002/rsa.21014

Let Pnsep denote the uniform probability measure on the set of separable permutations in Sn. Let ℕ∗=ℕ∪{∞} with an appropriate metric and denote by S(ℕ,ℕ∗) the compact metric space consisting of functions σ={σi}i=1∞ from ℕ to ℕ∗ which are injections when restricted to σ−1(ℕ); that is, if σi=σj, i ≠ j, then σi=∞. Extending permutations σ∈Sn
by defining σj=j, for j > n, we have Sn⊂S(ℕ,ℕ∗). We show that {Pnsep}n=1∞ converges weakly on S(ℕ,ℕ∗) to a limiting distribution of regenerative type, which we calculate explicitly.