LATIN 2016: LATIN 2016: Theoretical Informatics pp 509-521

# Unshuffling Permutations

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

## 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.

## Keywords

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.

## References

1. 1.
Allauzen, C.: Calcul efficace du shuffle de $$k$$ mots. Technical report, Institut Gaspard Monge, Université Marne-la-Vallée (2000)Google Scholar
2. 2.
Bose, P., Buss, J.F., Lubiw, A.: Pattern matching for permutations. Inf. Process. Lett. 65(5), 277–283 (1998)
3. 3.
Buss, S., Soltys, M.: Unshuffling a square is NP-hard. J. Comput. Syst. Sci. 80(4), 766–776 (2014)
4. 4.
Choffrut, C., Karhumäki, J.: Combinatorics of words. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages. Springer, Heidelberg (1997)Google Scholar
5. 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)
6. 6.
Eilenberg, S., Mac Lane, S.: On the groups of $$H(\Pi, n)$$. I. Ann. of Math. 58(2), 58:55–58:106 (1953)
7. 7.
Grinberg, D., Reiner, V.: Hopf Algebras in Combinatorics (2014). arxiv:1409.8356
8. 8.
Joni, S.A., Rota, G.-C.: Coalgebras and bialgebras in combinatorics. Stud. Appl. Math. 61(2), 93–139 (1979)
9. 9.
Mansfield, A.: On the computational complexity of a merge recognition problem. Discrete Appl. Math. 5, 119–122 (1983)
10. 10.
Henshall, D., Rampersad, N., Shallit, J.: Shuffling and unshuffling (2011). http://arxiv.org/abs/1106.5767
11. 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)
12. 12.
Simion, R., Schmidt, F.W.: Restricted permutations. Eur. J. Comb. 6(4), 383–406 (1985)
13. 13.
Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. https://oeis.org/
14. 14.
Spehner, J.-C.: Le calcul rapide des melanges de deux mots. Theoret. Comput. Sci. 47, 181–203 (1986)
15. 15.
van Leeuwen, J., Nivat, M.: Efficient recognition of rational relations. Inf. Process. Lett. 14(1), 34–38 (1982)
16. 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. 17.
Warmuth, M.K., Haussler, D.: On the complexity of iterated shuffle. J. Comput. Syst. Sci. 28(3), 345–358 (1984)