The Infinite limit of random permutations avoiding patterns of length three. (14th January 2020) - Pinsky, Ross G.
Abstract: For τ S_3, let μ _n^τ denote the uniformly random probability measure on the set of τ -avoiding 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^*) . For each τ S_3, we study the limiting behaviour of the measures { μ _n^τ } _n = 1^∞ on S(N, N^*) . We obtain partial results for the permutation τ = 321 and complete results for the other five permutations τ S_3 .