Unshuffling Permutations

  • Samuele GiraudoEmail author
  • Stéphane Vialette
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)


A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the problem of recognizing square permutations from both the point of view of algebra and algorithms. On the one hand, we present some algebraic and combinatorial properties of the shuffle product of permutations. We follow an unusual line consisting in defining the shuffle of permutations by means of an unshuffling operator, known as a coproduct. This strategy allows to obtain easy proofs for algebraic and combinatorial properties of our shuffle product. We besides exhibit a bijection between square (213, 231)-avoiding permutations and square binary words. On the other hand, by using a pattern avoidance criterion on oriented perfect matchings, we prove that recognizing square permutations is NP-complete.


Associative Algebra Combinatorial Property Linear Graph Multiple Occurrence Equivalent Manner 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Université Paris-Est, LIGM (UMR 8049), CNRS, UPEM, ESIEE Paris, ENPCMarne-la-ValléeFrance

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