Unshuffling Permutations: Trivial Bijections and Compositions

  • Guillaume Fertin
  • Samuele Giraudo
  • Sylvie Hamel
  • Stéphane VialetteEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11436)


Given permutations \(\pi \), \(\sigma _1\) and \(\sigma _2\), the permutation \(\pi \) (viewed as a string) is said to be a shuffle of \(\sigma _1\) and \(\sigma _2\), in symbols Open image in new window , if \(\pi \) can be formed by interleaving the letters of two strings \(p_1\) and \(p_2\) that are order-isomorphic to \(\sigma _1\) and \(\sigma _2\), respectively. Given a permutation \(\pi \in S_{2n}\) and a bijective mapping \(f : S_n \rightarrow S_n\), the f-Unshuffle-Permutation problem is to decide whether there exists a permutation \(\sigma \in S_n\) such that Open image in new window . We consider here this problem for the following bijective mappings: inversion, reverse, complementation, and all their possible compositions. In particular, we present combinatorial results about the permutations accepted by this problem. As main results, we obtain that this problem is \(\mathsf {NP}\)-complete when f is the reverse, the complementation, or the composition of the reverse with the complementation.


Permutation Shuffle product Computational complexity 


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Guillaume Fertin
    • 1
  • Samuele Giraudo
    • 2
  • Sylvie Hamel
    • 3
  • Stéphane Vialette
    • 2
    Email author
  1. 1.Université de Nantes, LS2N (UMR 6004), CNRSNantesFrance
  2. 2.Université Paris-Est, LIGM (UMR 8049), CNRS, ENPC, ESIEE Paris, UPEMMarne-la-ValléeFrance
  3. 3.Université de Montréal, DIROMontréalCanada

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