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Unshuffling Permutations

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LATIN 2016: Theoretical Informatics (LATIN 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9644))

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Abstract

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.

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References

  1. Allauzen, C.: Calcul efficace du shuffle de \(k\) mots. Technical report, Institut Gaspard Monge, Université Marne-la-Vallée (2000)

    Google Scholar 

  2. Bose, P., Buss, J.F., Lubiw, A.: Pattern matching for permutations. Inf. Process. Lett. 65(5), 277–283 (1998)

    Article  MathSciNet  Google Scholar 

  3. Buss, S., Soltys, M.: Unshuffling a square is NP-hard. J. Comput. Syst. Sci. 80(4), 766–776 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Choffrut, C., Karhumäki, J.: Combinatorics of words. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages. Springer, Heidelberg (1997)

    Google Scholar 

  5. Duchamp, G., Hivert, F., Thibon, J.-Y.: Noncommutative symmetric functions. VI. Free quasi-symmetric functions and related algebras. Int. J. Algebr. Comput. 12(5), 671–717 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Eilenberg, S., Mac Lane, S.: On the groups of \(H(\Pi, n)\). I. Ann. of Math. 58(2), 58:55–58:106 (1953)

    MathSciNet  Google Scholar 

  7. Grinberg, D., Reiner, V.: Hopf Algebras in Combinatorics (2014). arxiv:1409.8356

  8. Joni, S.A., Rota, G.-C.: Coalgebras and bialgebras in combinatorics. Stud. Appl. Math. 61(2), 93–139 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mansfield, A.: On the computational complexity of a merge recognition problem. Discrete Appl. Math. 5, 119–122 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Henshall, D., Rampersad, N., Shallit, J.: Shuffling and unshuffling (2011). http://arxiv.org/abs/1106.5767

  11. Rizzi, R., Vialette, S.: On recognizing words that are squares for the shuffle product. In: Bulatov, A.A., Shur, A.M. (eds.) CSR 2013. LNCS, vol. 7913, pp. 235–245. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  12. Simion, R., Schmidt, F.W.: Restricted permutations. Eur. J. Comb. 6(4), 383–406 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. https://oeis.org/

  14. Spehner, J.-C.: Le calcul rapide des melanges de deux mots. Theoret. Comput. Sci. 47, 181–203 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  15. van Leeuwen, J., Nivat, M.: Efficient recognition of rational relations. Inf. Process. Lett. 14(1), 34–38 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  16. Y. Vargas. Hopf algebra of permutation pattern functions. In: 26th International Conference on Formal Power Series and Algebraic Combinatorics, pp. 839–850 (2014)

    Google Scholar 

  17. Warmuth, M.K., Haussler, D.: On the complexity of iterated shuffle. J. Comput. Syst. Sci. 28(3), 345–358 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Université Paris-Est, LIGM (UMR 8049), CNRS, UPEM, ESIEE Paris, ENPC, 77454, Marne-la-Vallée, France

    Samuele Giraudo & Stéphane Vialette

Authors
  1. Samuele Giraudo
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  2. Stéphane Vialette
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Correspondence to Samuele Giraudo .

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Editors and Affiliations

  1. Carleton University, Ottawa, ON, Ontario, Canada

    Evangelos Kranakis

  2. University Chile, Santiago, Chile

    Gonzalo Navarro

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Giraudo, S., Vialette, S. (2016). Unshuffling Permutations. In: Kranakis, E., Navarro, G., Chávez, E. (eds) LATIN 2016: Theoretical Informatics. LATIN 2016. Lecture Notes in Computer Science(), vol 9644. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49529-2_38

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  • DOI: https://doi.org/10.1007/978-3-662-49529-2_38

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-49528-5

  • Online ISBN: 978-3-662-49529-2

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