Bijective Construction of Equivalent Eco-systems

  • Srečko Brlek
  • Enrica Duchi
  • Elisa Pergola
  • Renzo Pinzani
Conference paper
Part of the Trends in Mathematics book series (TM)


First we explicit an infinite family of equivalent succession rules parametrized by a positive integer α, for which two specializations lead to the equivalence of rules defining the Catalan and Schröder numbers. Then, from an ECO-system for Dyck paths easily derive an ECO-system for complete binary trees y using a widely known bijection between these objects. We also give a similar construction in the less easy case of Schröder paths and Schröder trees which generalizes the previous one.


Lattice Path Easy Case Combinatorial Object Succession Rule Type Path 
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  1. [1]
    C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy, and D. Gouyou-Beauchamps, (2002)Generating functions for generating treesDiscrete Mathematics, 246 (1–3) 29–55.zbMATHCrossRefGoogle Scholar
  2. [2]
    E. Barcucci, A. Del Lungo, E. Pergola, R. Pinzani, (1999)ECO: a methodology for the Enumeration of Combinatorial ObjectsJournal of Difference Equations and Applications, Vol.5, 435–490.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    M. Bousquet-Mélou, M. Petkovgek, (2000)Linear Recurrences with Constant Coefficients: the Multivariate Case, Discrete Mathematics225 (1–3)51–75.zbMATHGoogle Scholar
  4. [4]
    F. R. K. Chung, R. L. Graham, V. E. Hoggatt, M. Kleimann, (1978)The number of Baxter permutations, Journal of Combinatorial Theory Ser. A24, 382–394.zbMATHCrossRefGoogle Scholar
  5. [5]
    S. Corteel, (2000) Séries génératrices exponentielles pour les eco-systèmes signés, in D. Krob, A.A. Mikhalev, A.V. Mikhalev (Eds.)Proceedings of the 12-th International Conference on Formal Power Series and Algebraic Combinatorics Moscow RussiaSpringer, 655–666.Google Scholar
  6. [6]
    S. Gire, (1993)Arbres permutations à motifs exclus et cartes planaires: quelques problèmes algorithmiques et combinatoiresThèse de l’université de Bordeaux I.Google Scholar
  7. [7]
    O. Guibert, (1995)Combinatoire des permutations à motifs exclus en liaison avec mots cartes planaires et tableaux de YoungThèse de l’université de Bordeaux I.Google Scholar
  8. [8]
    J.G. Penaud, E. Pergola, R. Pinzani, O. Roques, (2001)Chemins de Schröder et hiérarchies aléatoires, Theoretical Computer Science255, 345–361.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    E. Pergola, R. Pinzani, S. Rinaldi, (2000)A set of well-defined operations on succession rulesProceedings of MCS, Versailles,l, 141–152.Google Scholar
  10. [10]
    R. P. Stanley, (1986)Enumerative Combinatorics Vol. IWadworth & Brooks/Cole, Monterey, Cal.Google Scholar
  11. [11]
    R. A. Sulanke, (2000)Moments of Generalized Motzkin Paths, J. of Integer SequencesVol. 3, article 00.1.1.Google Scholar
  12. [12]
    J. West, (1995)Generating trees and the Catalan and Schröder numbers, Discrete Mathematics146, 247–262.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Srečko Brlek
    • 1
  • Enrica Duchi
    • 2
  • Elisa Pergola
    • 2
  • Renzo Pinzani
    • 2
  1. 1.LaCIMUniversité du Québec à MontréalCentre-Ville, Montréal (QC)Canada
  2. 2.Dipartimento di Sistemi e InformaticaFirenzeItaly

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