Pattern Matching for 321-Avoiding Permutations

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Given two permutations π and σ, the NP-complete Permutation Pattern problem is to decide whether π contains σ as a pattern. In case both π and σ are 321-avoiding, we prove the Permutation Pattern problem to be solvable in O(k 2 n 6) time, where k = |σ| and n = |π|, and give a \(O(kn^{4\sqrt{k}+12})\) time algorithm if only σ is 321-avoiding. Finally, we show W[1]-hardness of a 2-colored version of this latter problem.