For Mallows(q) permutations, the probability of avoiding pattern 123 decays as q^(n^2/4); avoiding 132 or 213 decays as (1-q)^n; avoiding 312 or 231 has a rate characterized by a functional equation.
and Peled, R., Lengths of monotone subsequences in a Mallows per- mutation, Probab
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Permutations avoiding a pattern of length three under Mallows distributions
For Mallows(q) permutations, the probability of avoiding pattern 123 decays as q^(n^2/4); avoiding 132 or 213 decays as (1-q)^n; avoiding 312 or 231 has a rate characterized by a functional equation.