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Tight Multiple Twins in Permutations


Two permutations are similar if they have the same length and the same relative order. A collection of \(r\geqslant 2\) disjoint, similar subsequences of a permutation \(\pi \) forms r-twins in \(\pi \). We study the longest guaranteed length of r-twins which are tight in the sense that either each twin alone forms a block or their union does. We address the same question with respect to a random permutation.

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  1. 1.

    N. Alon, J. Spencer, The Probabilistic Method, Fourth Edition, Wiley, 2016.

  2. 2.

    S. V. Avgustinovich, S. Kitaev, A. Pyatkin, A. Valyuzhenich, On square-free permutations, J. Autom. Lang. Comb. 16 (2011) 3–10.

    MathSciNet  MATH  Google Scholar 

  3. 3.

    B. Bukh and O. Rudenko, Order-isomorphic twins in permutations, SIAM J. Discrete Math. 34 (2020), no. 3, 1620–1622.

    MathSciNet  Article  Google Scholar 

  4. 4.

    A. Dudek, J. Grytczuk and A. Ruciński, Variations on twins in permutations, Electronic Journal of Combinatorics 28 (2021), no. 3, #P3.19.

    MathSciNet  Article  Google Scholar 

  5. 5.

    A. Dudek, J. Grytczuk and A. Ruciński, On weak twins and up-and-down sub-permutations, Integers 21A (2021), Ron Graham Memorial Volume, Paper No. A10, 17 pp.

  6. 6.

    P. Erdős, L. Lovász, Problems and results on 3-chromatic hypergraphs and some related questions, in: Infinite and Finite Sets (A. Hajnal et al., eds.), North-Holland, Amsterdam, (1975) 609–628.

    Google Scholar 

  7. 7.

    A. Frieze and B. Pittel, Perfect matchings in random graphs with prescribed minimal degree. Mathematics and computer science. III, 95–132, Trends Math., Birkhäuser, Basel, 2004.

  8. 8.

    M. Gawron, Izomorficzne podstruktury w słowach i permutacjach, Master Thesis (in Polish), 2014.

  9. 9.

    M. Lothaire, Combinatorics on words, Addison-Wesley, Reading, MA, 1983.

    MATH  Google Scholar 

  10. 10.

    C. McDiarmid, Concentration. Probabilistic methods for algorithmic discrete mathematics, 195–248, Algorithms Combin., 16, Springer, Berlin, 1998.

  11. 11.

    C. Schine, The Grammarians, Sarah Crichton Books; First Edition (September 3, 2019).

  12. 12.

    A. Thue, Über unendliche Zeichenreichen, Norske Vid. Selsk. Skr., I Mat. Nat. Kl., Christiania 7 (1906) 1–22.

    Google Scholar 

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We would like to thank an anonymous referee for a careful reading of the manuscript and suggesting a number of editorial improvements.

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Correspondence to Andrzej Dudek.

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Andrzej Dudek was supported in part by Simons Foundation Grant #522400. Jarosław Grytczuk was supported in part by Narodowe Centrum Nauki, grant 2015/17/B/ST1/02660. Andrzej Ruciński was supported in part by Narodowe Centrum Nauki, grant 2018/29/B/ST1/00426.

Communicated by Matjaz Konvalinka.

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Dudek, A., Grytczuk, J. & Ruciński, A. Tight Multiple Twins in Permutations. Ann. Comb. (2021).

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