The Infinite limit of random permutations avoiding patterns of length three

For tau is an element of S3, let mu(n)(t) denote the uniformly random probability measure on the set of t -avoiding permutations in S-n. Let N* = N boolean OR {infinity} with an appropriate metric and denote by S(N, N*) the compact metric space consisting of functions sigma = {sigma(i)}(i=1)(infinity) from N to N* which are injections when restricted to sigma(-1)(N); that is, if sigma(i) = sigma(j), i not equal j, then sigma(i) = infinity. Extending permutations sigma is an element of S-n by defining sigma(j) = j, for j > n, we have S-n subset of S(N, N*). For each tau is an element of S-3, we study the limiting behaviour of the measures {mu(t)(n)}(n=1)(infinity) on S(N, N*). We obtain partial results for the permutation tau = 321 and complete results for the other five permutations tau is an element of S-3.