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A permutation is said to be a square if it can be obtained by shuffling twoorder-isomorphic patterns. The definition is intended to be the naturalcounterpart to the ordinary shuffle of words and languages. In this paper, wetackle the problem of recognizing square permutations from both the point ofview of algebra and algorithms. On the one hand, we present some algebraic andcombinatorial properties of the shuffle product of permutations. We follow anunusual line consisting in defining the shuffle of permutations by means of anunshuffling operator, known as a coproduct. This strategy allows to obtain easyproofs for algebraic and combinatorial properties of our shuffle product. Webesides exhibit a bijection between square $(213,231)$-avoiding permutationsand square binary words. On the other hand, by using a pattern avoidancecriterion on directed perfect matchings, we prove that recognizing squarepermutations is {\bf NP}-complete.

Algorithmic and algebraic aspects of unshuffling permutations