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Treelike Structures

  • Samuele Giraudo
Chapter

Abstract

This second chapter is devoted to present general notions about treelike structures. We present more precisely the ones appearing in the algebraic and combinatorial context of nonsymmetric operads. Rewrite systems of syntax trees are exposed, as well as methods to prove their termination and their confluence.

References

  1. [BD16]
    M.R. Bremner, V. Dotsenko, Algebraic Operads: An Algorithmic Companion (Chapman and Hall/CRC, London/Boca Raton, 2016), pp. xvii+365Google Scholar
  2. [Cay57]
    A. Cayley, On the theory of the analytical forms called trees. Philos. Mag. 13, 172–176 (1857)CrossRefGoogle Scholar
  3. [CCG18]
    C. Chenavier, C. Cordero, S. Giraudo, Generalizations of the associative operad and convergent rewrite systems. Higher Dimens. Rewriting Algebra (2018). https://doi.org/10.29007/mfnh
  4. [Cha08]
    F. Chapoton, Operads and algebraic combinatorics of trees. Sém. Lothar. Combin. B58c, 27 (2008)Google Scholar
  5. [CLRS09]
    T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms, 3rd edn. (The MIT Press, Cambridge, 2009)zbMATHGoogle Scholar
  6. [DK10]
    V. Dotsenko, A. Khoroshkin, Gröbner bases for operads. Duke Math. J. 153(2), 363–396 (2010)MathSciNetCrossRefGoogle Scholar
  7. [DM47]
    A. Dvoretzky, Th. Motzkin, A problem of arrangements. Duke Math. J. 14(2), 305–313 (1947)MathSciNetCrossRefGoogle Scholar
  8. [EFM14]
    K. Ebrahimi-Fard, D. Manchon, The Magnus expansion, trees and Knuth’s rotation correspondence. Found. Comput. Math. 14(1), 1–25 (2014)MathSciNetCrossRefGoogle Scholar
  9. [FS09]
    P. Flajolet, R. Sedgewick, Analytic Combinatorics (Cambridge University Press, Cambridge, 2009)CrossRefGoogle Scholar
  10. [Gir16b]
    S. Giraudo, Operads from posets and Koszul duality. Eur. J. Comb. 56C, 1–32 (2016)MathSciNetCrossRefGoogle Scholar
  11. [HNT05]
    F. Hivert, J.-C. Novelli, J.-Y. Thibon, The algebra of binary search trees. Theor. Comput. Sci. 339(1), 129–165 (2005)MathSciNetCrossRefGoogle Scholar
  12. [Hof10]
    E. Hoffbeck, A Poincaré-Birkhoff-Witt criterion for Koszul operads. Manuscripta Math. 131(1-2), 87–110 (2010)MathSciNetCrossRefGoogle Scholar
  13. [Knu97]
    D. Knuth, The Art of Computer Programming. Fundamental Algorithms, 3rd edn., vol. 1 (Addison Wesley Longman, Boston, 1997), pp. xx+650Google Scholar
  14. [Knu98]
    D. Knuth, The Art of Computer Programming. Sorting and Searching, vol. 3 (Addison Wesley Longman, Boston, 1998), pp. xiv+780Google Scholar
  15. [Lab81]
    G. Labelle, Une nouvelle démonstration combinatoire des formules d’inversion de Lagrange. Adv. Math. 42(3), 217–247 (1981)MathSciNetCrossRefGoogle Scholar
  16. [LR98]
    J.-L. Loday, M. Ronco, Hopf algebra of the planar binary trees. Adv. Math. 139, 293–309 (1998)MathSciNetCrossRefGoogle Scholar
  17. [LV12]
    J.-L. Loday, B. Vallette, Algebraic Operads. Grundlehren der mathematischen Wissenschaften, vol. 346 (Springer, Heidelberg, 2012), pp. xxiv+634CrossRefGoogle Scholar
  18. [Nar55]
    T.V. Narayana, Sur les treillis formés par les partitions d’un entier et leurs applications à la théorie des probabilités. C. R. Acad. Sci. Paris 240, 1188–1189 (1955)MathSciNetzbMATHGoogle Scholar
  19. [NT13]
    J.-C. Novelli, J.-Y. Thibon, Duplicial algebras and Lagrange inversion (2013). arXiv:1209.5959v2
  20. [Slo]
    N.J.A. Sloane, The on-line encyclopedia of integer sequences (1996). https://oeis.org/

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Samuele Giraudo
    • 1
  1. 1.University of Paris-EstMarne-la-ValleeFrance

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