Analytic Combinatorics of Chord Diagrams

  • Philippe Flajolet
  • Marc Noy
Conference paper


In this paper we study the enumeration of diagrams of n chords joining 2n points on a circle in disjoint pairs. We establish limit laws for the following three parameters: number of components, size of the largest component, and number of crossings. We also find exact formulas for the moments of the distribution of number of components and number of crossings.


Cayley Tree Large Connected Component Chord Diagram Link Diagram Bell Polynomial 
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  1. 1.
    P. Billingsley, Probability and measure, third edition, John Wiley and Sons (1995).Google Scholar
  2. 2.
    L. Comtet, Advanced Combinatorics, Reidel (1974).Google Scholar
  3. 3.
    N.G. De Bruijn, Asymptotic Methods in Analysis, Dover (1981).Google Scholar
  4. 4.
    P. Flajolet, Combinatorial aspects of continued fractions, Discrete Math. 32 (1980), 125 - 161.MathSciNetMATHGoogle Scholar
  5. 5.
    P. Flajolet, J. Françon and J. Vuillemin, Sequence of operations analysis for dynamic data structures, J. Algorithms 1 (1980), 111 - 141.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    P. Flajolet and M. Noy, Analytic Combinatorics of Non-crossing Configurations, Discrete Mathematics 204 (1999), 203 - 229.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    P. Flajolet, C. Puech and J. Vuillemin, The analysis of simple list structures, Inform. Sci. 38 (1986), 121 - 146.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    P. Flajolet and B. Salvy, unpublished manuscript.Google Scholar
  9. 9.
    P. Flajolet and R. Sedgewick, Analytic Combinatorics, book in preparation (individual chapters are available as INRIA Research Reports 1888, 2026, 2376, 2956, 3162 ).Google Scholar
  10. 10.
    M.E.H. Ismail, D. Stanton and G. Viennot, The combinatorics of q-Hermite polynomials and the Askey-Wilson integral, European J. Combin. 8 (1987), 379 - 392.MathSciNetMATHGoogle Scholar
  11. 11.
    A. Nijenhuis and H.S. Wilf, The Enumeration of Connected Graphs and Linked Diagrams, J. Combinatorial Theory A 27 (1979), 356 - 359.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    J.-G. Penaud, Une preuve bijective d'une formule de Touchard-Riordan, Discrete Mathematics 139 (1995), 347 - 360.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    R.C. Read, The chord intersection problem Annals of the New York Academy of Sciences 139 (1979), 444-454Google Scholar
  14. 14.
    J. Riordan, The Distribution of Crossings of Chords Joining Pairs of 2n points on a Circle, Math. of Computation 29 (1975), 215 - 222.MathSciNetMATHGoogle Scholar
  15. 15.
    R. Sedgewick and P. Flajolet, An introduction to the analysis of algorithms,Addison-Wesley (1996)Google Scholar
  16. 16.
    P.R. Stein, On a Class of Linked Diagrams, I. Enumeration J. of Combinatorial Theory A 24 (1978), 357-366Google Scholar
  17. 17.
    P.R. Stein and C.J. Everett, On a Class of Linked Diagrams, II. Asymptotics Discrete Mathematics 21 (1978), 309-318Google Scholar
  18. 18.
    A. Stoimenow, Enumeration of chord diagrams and an upper bound for Vassiliev invariants, J. Knot Theory Ramifications 7 (1998),93-114.Google Scholar
  19. 19.
    J. Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math. 4 (1952), 2 - 25.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Philippe Flajolet
    • 1
  • Marc Noy
    • 2
  1. 1.INRIARocquencourtFrance
  2. 2.Universitat Politècnica de CatalunyaBarcelonaSpain

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