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Given two permutations, a pattern σ and a text π, Parity Permutation Pattern Matching asks whether there exists a parity and order preserving embedding of σ into π. While it is known that Permutation Pattern Matching is in FPT, we show that adding the parity constraint to the problem makes it W[1]-hard, even for alternating permutations or for 4321-avoiding patterns. However, it remains in FPT if the text avoids a fixed permutation, thanks to a recent meta-theorem on twin-width. On the other hand, as for the classical version, Parity Permutation Pattern Matching remains polynomial-time solvable when both permutations are separable, or if both are 321-avoiding, but NP-hard if the pattern is 321-avoiding and the text is 4321-avoiding.

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