Increasing Diamonds

  • Olivier Bodini
  • Matthieu Dien
  • Xavier Fontaine
  • Antoine GenitriniEmail author
  • Hsien-Kuei Hwang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)


A class of diamond-shaped combinatorial structures is studied whose enumerating generating functions satisfy differential equations of the form \(f'' = G(f)\), for some function G. In addition to their own interests and being natural extensions of increasing trees, the study of such DAG-structures was motivated by modelling executions of series-parallel concurrent processes; they may also be used in other digraph contexts having simultaneously a source and a sink, and are closely connected to a few other known combinatorial structures such as trees, cacti and permutations. We explore in this extended abstract the analytic-combinatorial aspect of these structures, as well as the algorithmic issues for efficiently generating random instances.


  1. 1.
    Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (2012)zbMATHGoogle Scholar
  2. 2.
    Ando, E., Nakata, T., Yamashita, M.: Approximating the longest path length of a stochastic DAG by a normal distribution in linear time. J. Discrete Algorithms 7(4), 420–438 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bergeron, F., Flajolet, P., Salvy, B.: Varieties of increasing trees. In: Raoult, J.-C. (ed.) CAAP ’92. LNCS, vol. 581, pp. 24–48. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  4. 4.
    Bodini, O.: Autour de la génération aléatoire sous modèle de Boltzmann. Habilitation thesis, UPMC (2010)Google Scholar
  5. 5.
    Bodini, O., Roussel, O., Soria, M.: Boltzmann samplers for first-order differential specifications. Discrete Appl. Math. 160(18), 2563–2572 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chern, H.-H., Fernández-Camacho, M.-I., Hwang, H.-K., Martínez, C.: Psi-series method for equality of random trees and quadratic convolution recurrences. Random Struct. Algorithms 44(1), 67–108 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Duchon, P., Flajolet, P., Louchard, G., Schaeffer, G.: Boltzmann samplers for the random generation of combinatorial structures. Comb. Prob. Comput. 13(4–5), 577–625 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  9. 9.
    Knuth, D.E.: The Art of Computer Programming, volume 1 (3rd ed.): Fundamental Algorithms, Addison Wesley Longman Publishing Co., Inc., Redwood City, CA, USA (1997)Google Scholar
  10. 10.
    Kuba, M., Panholzer, A.: A combinatorial approach to the analysis of bucket recursive trees. Theor. Comput. Sci. 411(34–36), 3255–3273 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kuba, M., Panholzer, A.: Bilabelled increasing trees and hook-length formulae. Eur. J. Combin. 33(2), 248–258 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kuba, M., Panholzer, A.: Combinatorial families of multilabelled increasing trees and hook-length formulas. Discrete Math. 339, 227–254 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Meir, A., Moon, J.W.: On the altitude of nodes in random trees. Can. J. Math. 30(5), 997–1015 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Stanley, R.: Catalan Numbers. Cambridge University Press, Cambridge (2015)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Olivier Bodini
    • 1
  • Matthieu Dien
    • 2
  • Xavier Fontaine
    • 1
  • Antoine Genitrini
    • 2
    Email author
  • Hsien-Kuei Hwang
    • 3
  1. 1.Laboratoire d’Informatique de Paris-NordCNRS UMR 7030 - Institut Galilée - Université Paris-NordVilletaneuseFrance
  2. 2.Sorbonne Universités, UPMC Univ Paris 06, CNRS, LIP6 UMR 7606ParisFrance
  3. 3.Institute of Statistical ScienceAcademia SinicaTaipeiTaiwan

Personalised recommendations