Clustering in Random Pattern-Avoiding Permutations

A cluster of length l in a permutation from Sn is a set of l consecutive numbers that appear in any order in l consecutive positions in the permutation. For n ≥ l ≥ 2 and τ ∈ S3, let N(n) l (σ) denote the number of clusters of length l in σ, and let N(n) l;τ (σ) denote the number of clusters of length l whose order is the pattern τ . (For example, σ = 375619248 ∈ S9 has the cluster 756 of length l = 3 and pattern τ = 213.)