The patterns of permutations

• Authors: Herbert S. Wilf
• Venue: Discrete Mathematics - Kleitman and combinatorics: a celebration
• Year: 2002

Let n,k be positive integers, with k ≤ n, and let τ be a fixed permutation of {1,...,k}. We will call τ the pattern. We will look for the pattern τ in permutations σ of n letters. A pattern τ is said to occur in a permutation σ if there are integers 1 ≤ i1 i2 ... ik ≤ n such that for all 1 ≤ r s ≤ k we have τ(r) τ(s) if and only if σ(ir) σ(is).Example. Suppose τ = (132). Then this pattern of k = 3 letters occurs several times in the following permutation σ, of n = 14 letters (one such occurrence is underlined): σ=(5 2 9 4 14 10 1 3 6 15 8 11 7 13 12).