Clustering of consecutive numbers in permutations avoiding a pattern of length three or avoiding a finite number of simple patterns

For η∈S₃, let S_n^av(η) denote the set of permutations in S_n that avoid the pattern η, and let E_n^av(η) denote the expectation with respect to the uniform probability measure on S_n^av(η). For n≥k≥2 and τ∈S_k^av(η), let N_n^(k)(σ) denote the number of occurrences of k consecutive numbers appearing in k consecutive positions in σ∈S_n^av(η), and let N_n^(k;τ)(σ) denote the number of such occurrences for which the order of the appearance of the k numbers is the pattern τ. We obtain explicit formulas for E_n^av(η)N_n^(k;τ) and E_n^av(η)N_n^(k), for all 2≤k≤n, all η∈S₃ and all τ∈S_k^av(η). These exact formulas then yield asymptotic formulas as n→∞ with k fixed, and as n→∞ with k=k_n→∞. We also obtain analogous results for S_n^{av(η₁,⋯,η_r)}, the subset of S_n consisting of permutations avoiding the patterns {η_i}_i=1^r, where η_i∈S_mi, in the case that {η_i}_i=1ⁿ are all simple permutations. A particular case of this is the set of separable permutations, which corresponds to r=2, η₁=2413,η₂=3142.