Enriched Collections

  • Samuele Giraudo


This preliminary chapter contains general notions about combinatorics used in the rest of the book. We introduce the notion of collections of combinatorial objects and then the notions of posets and rewrite systems, which are seen as collections endowed with some extra structure.


  1. [AVL62]
    G.M. Adelson-Velsky, E.M. Landis, An algorithm for the organization of information. Sov. Math. Dokl. 3, 1259–1263 (1962)Google Scholar
  2. [Bjö84]
    A Björner, Orderings of coxeter groups, in Combinatorics and Algebra. Contemporary Mathematics, vol. 34 (American Mathematical Society, Providence, 1984), pp. 175–195Google Scholar
  3. [BLL98]
    F. Bergeron, G. Labelle, P. Leroux, Combinatorial Species and Tree-Like Structures. Encyclopedia of Mathematics and Its Applications, vol. 67 (Cambridge University Press, Cambridge, 1998)Google Scholar
  4. [BLL13]
    F. Bergeron, G. Labelle, P. Leroux, Introduction to the Theory of Species of Structures (Université du Québec à Montréal, Montreal, 2013)Google Scholar
  5. [BMFPR11]
    M. Bousquet-Mélou, É. Fusy, L.-F. Préville-Ratelle, The number of intervals in the m-Tamari lattices. Electron. J. Comb. 18(2) (2011); Paper 31Google Scholar
  6. [BN98]
    F. Baader, T. Nipkow, Term Rewriting and All That (Cambridge University Press, Cambridge, 1998), pp. xii+301Google Scholar
  7. [BPR12]
    F. Bergeron, L.-F. Préville-Ratelle, Higher trivariate diagonal harmonics via generalized Tamari posets. J. Comb. 3(3), 317–341 (2012)MathSciNetzbMATHGoogle Scholar
  8. [Buc76]
    B. Buchberger, A theoretical basis for the reduction of polynomials to canonical forms. ACM SIGSAM Bull. 10(3), 19–29 (1976)MathSciNetCrossRefGoogle Scholar
  9. [Cha06]
    F. Chapoton, Sur le nombre d’intervalles dans les treillis de Tamari. Sém. Lothar. Combin. B55f, 18 (2006)Google Scholar
  10. [Cox34]
    H.S.M. Coxeter, Discrete groups generated by reflections. Ann. Math. 35(3), 588–621 (1934)MathSciNetCrossRefGoogle Scholar
  11. [FS09]
    P. Flajolet, R. Sedgewick, Analytic Combinatorics (Cambridge University Press, Cambridge, 2009)CrossRefGoogle Scholar
  12. [GR63]
    G.Th. Guilbaud, P. Rosenstiehl, Analyse algébrique d’un scrutin. Math. Sci. Hum. 4, 9–33 (1963)Google Scholar
  13. [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
  14. [HT72]
    S. Huang, D. Tamari, Problems of associativity: a simple proof for the lattice property of systems ordered by a semi-associative law. J. Comb. Theory A 13, 7–13 (1972)MathSciNetCrossRefGoogle Scholar
  15. [Joy81]
    A. Joyal, Une théorie combinatoire des séries formelles. Adv. Math. 42(1), 1–82 (1981)MathSciNetCrossRefGoogle Scholar
  16. [KB70]
    D. Knuth, P. Bendix, Simple word problems in universal algebras, in Computational Problems in Abstract Algebra (Pergamon, Oxford, 1970), pp. 263–297zbMATHGoogle Scholar
  17. [Knu98]
    D. Knuth, The Art of Computer Programming. Sorting and Searching, vol. 3 (Addison Wesley Longman, Boston, 1998), pp. xiv+780Google Scholar
  18. [Koc09]
    J. Kock, Notes on Polynomial Functors. Unpublished (2009).
  19. [LR98]
    J.-L. Loday, M. Ronco, Hopf algebra of the planar binary trees. Adv. Math. 139, 293–309 (1998)MathSciNetCrossRefGoogle Scholar
  20. [LR02]
    J.-L. Loday, M. Ronco, Order structure on the algebra of permutations and of planar binary trees. J. Algebra Comb. 15(3), 253–270 (2002)MathSciNetCrossRefGoogle Scholar
  21. [Mén15]
    M.A. Méndez, Set Operads in Combinatorics and Computer Science. SpringerBriefs in Mathematics (Springer, Cham, 2015), pp. xvi+129Google Scholar
  22. [New42]
    M.H.A. Newman, On theories with a combinatorial definition of “equivalence”. Ann. Math. 43(2), 223–243 (1942)MathSciNetCrossRefGoogle Scholar
  23. [Slo]
    N.J.A. Sloane, The on-line encyclopedia of integer sequences (1996).
  24. [Tam62]
    D. Tamari, The algebra of bracketings and their enumeration. Nieuw Arch. Wisk. 10(3), 131–146 (1962)MathSciNetzbMATHGoogle Scholar
  25. [YO69]
    T. Yanagimoto, M. Okamoto, Partial orderings of permutations and monotonicity of a rank correlation statistic. Ann. I. Stat. Math. 21, 489–506 (1969)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

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

Personalised recommendations