Permutation patterns and statistics

Call two sequences of distinct integers a1a2 . . . ak and b1b2 . . . bk order isomorphic if they have the same pairwise comparisons, i.e., ai < aj if and only if bi < bj for all indices i, j. For example 132 and 475 are order isomorphic since both begin with the smallest element, have the largest element second, and end with the middle sized element. Let Sn be the set of all permutations of {1, 2, . . . , n} viewed as sequences π = a1a2 . . . an. We say that σ ∈ Sn contains π ∈ Sk as a pattern if there is a subsequence σ′ of σ which is order isomorphic to π. To illustrate, σ = 6473521 contains π = 132 as a pattern because of the subsequence σ′ = 475. If σ does not contain π, we say it avoids π and use the notation Avn(π) = {σ ∈ Sn : σ avoids π}.