# Unshuffling Permutations: Trivial Bijections and Compositions

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

## Abstract

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 , 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 . 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.

## Keywords

Permutation Shuffle product Computational complexity

## References

1. 1.
Bose, P., Buss, J.F., Lubiw, A.: Pattern matching for permutations. Inf. Process. Lett. 65(5), 277–283 (1998)
2. 2.
Buss, S., Soltys, M.: Unshuffling a square is NP-hard. J. Comput. Syst. Sci. 80(4), 766–776 (2014)
3. 3.
Choffrut, C., Karhumäki, J.: Combinatorics of words. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, pp. 329–438. Springer, Heidelberg (1997).
4. 4.
Giraudo, S., Vialette, S.: Algorithmic and algebraic aspects of unshuffling permutations. Theor. Comput. Sci. 729, 20–41 (2018)
5. 5.
Knuth, D.E.: The Art of Computer Programming: Volume III: Sorting and Searching. Addison-Wesley, Boston (1973)
6. 6.
van Leeuwen, J., Nivat, M.: Efficient recognition of rational relations. Inf. Process. Lett. 14(1), 34–38 (1982)
7. 7.
Mansfield, A.: On the computational complexity of a merge recognition problem. Discrete Appl. Math. 5(1), 119–122 (1983)
8. 8.
Neou, B.E., Rizzi, R., Vialette, S.: Pattern matching for separable permutations. In: Inenaga, S., Sadakane, K., Sakai, T. (eds.) SPIRE 2016. LNCS, vol. 9954, pp. 260–272. Springer, Cham (2016).
9. 9.
Rizzi, R., Vialette, S.: On recognizing words that are squares for the shuffle product. In: Proceedings of the 8th International Symposium in Computer Science - Theory and Applications, pp. 235–245 (2013)Google Scholar
10. 10.
Sloane, N.J.A.: The on-line encyclopedia of integer sequences. https://oeis.org/
11. 11.
Stankova, Z.: Forbidden subsequences. Discrete Math. 132(1–3), 291–316 (1994)
12. 12.
Vargas, Y.: Hopf algebra of permutation pattern functions. In: 26th International Conference on Formal Power Series and Algebraic Combinatorics, pp. 839–850 (2014)Google Scholar

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## 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