Generalized Pattern Frequency in Large Permutations

In the study of permutations, generalized patterns extend classical patterns by allowing the requirement that certain adjacent integers in a pattern must be adjacent in the permutation. For any generalized pattern π∗ 0 of length k with 1 ≤ b ≤ k blocks, we prove that for all μ > 0, there exists 0 < c = c(k, μ) < 1 so that whenever n ≥ n0(k, μ, c), all but cnn! many π ∈ Sn admit (1± μ) 1 k! (n b ) occurrences of π∗ 0 . Up to the choice of c, this result is best possible. We also give a lower bound on avoidance of the generalized pattern 12-34, which answers a question of S. Elizalde [8].