Basic concepts

  • Christian Reidys


In this chapter we provide the mathematical foundation for the following results. One main objective here is the self-contained derivation of the generating function of k-noncrossing matchings, which will play a central role for RNA pseudoknot structures.


Cayley Graph Vertex Boundary Weyl Chamber Combinatorial Class Reflection Principle 
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  1. 1.
    M. Abramowitz and I.A. Stegun, editors. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 55. NBS Applied Mathematics, Dover, NY 1964.Google Scholar
  2. 2.
    M. Ajtai, J. Komlós, and E. Szemerédi. Largest random component of a k-cube. Combinatorica, 2:1–7, 1982.MATHCrossRefMathSciNetGoogle Scholar
  3. 4.
    D. Aldous and P. Diaconis. Strong uniform times and finite random walks. Adv. Appl. Math., 2:69–97, 1987.CrossRefMathSciNetGoogle Scholar
  4. 5.
    D. André. Solution directe du problème résolu par M. Bertrand,. C R d Acad Sci, 105:436–437, 1887.Google Scholar
  5. 6.
    L. Babai. Local expansion of vertex transitive graphs and random generation in finite groups. Proc 23 ACM Symp Theory Comput (ACM New York), 1:164–174, 1991.Google Scholar
  6. 7.
    L. Babai and V.T. Sos. Sidon sets in groups and induced subgraphs of Cayley graphs. Eur. J. Combinator., 1:1–11, 1985.MathSciNetGoogle Scholar
  7. 14.
    B. Bollobás, Y. Kohayakawa, and T. Luczak. The evolution of random subgraphs of the cube. Random Struct. Algorithms, 3:55–90, 1992.MATHCrossRefGoogle Scholar
  8. 25.
    W.Y.C. Chen, E.Y.P. Deng, R.R.X. Du, R.P. Stanley, and C.H. Yan. Crossing and nesting of matchings and partitions. Trans. Amer. Math. Soc., 359:1555–1575, 2007.MATHCrossRefMathSciNetGoogle Scholar
  9. 27.
    W.Y.C. Chen, J. Qin, and C.M. Reidys. Crossing and nesting in tangled-diagrams. Elec. J. Comb., 15, 86, 2008.MathSciNetGoogle Scholar
  10. 36.
    G. P. Egorychev. Integral Representation and the computation of combinatorial sums, volume 59. American Mathematical Society, NY, 1984.Google Scholar
  11. 41.
    P. Flajolet, J.A. Fill, and N. Kapur. Singularity analysis, hadamard products, and tree recurrences. J. Comp. Appl. Math., 174:271–313, 2005.MATHCrossRefMathSciNetGoogle Scholar
  12. 42.
    P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge University Press, Cambridge, England, 2009.Google Scholar
  13. 49.
    I.M. Gessel and D. Zeilberger. Random walk in a Weyl chamber. Proc. Am. Math. Soc., 115:27–31, 1992.MATHMathSciNetGoogle Scholar
  14. 53.
    D.J. Grabiner and P. Magyar. Random walks in Weyl chambers and the decomposition of tensor powers. J. Algebr. Combinator., 2:239–260, 1993.MATHCrossRefMathSciNetGoogle Scholar
  15. 54.
    L.C. Grove and C.T. Benson. Finite reflection groups. Springer, New York, 1985.Google Scholar
  16. 58.
    Chernoff. H. A measure of the asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Stat., 23:493–509, 1952.MATHCrossRefMathSciNetGoogle Scholar
  17. 62.
    T.E. Harris. The Theory of Branching Processes. Springer, 1963.Google Scholar
  18. 66.
    P. Henrici. Applied and Computational Complex Analysis, volume 2. John Wiley, 1974.Google Scholar
  19. 69.
    I.L. Hofacker, P. Schuster, and P.F. Stadler. Combinatorics of RNA secondary structures. Discr. Appl. Math., 88:207–237, 1998.MATHCrossRefMathSciNetGoogle Scholar
  20. 74.
    N. Iwahori. On the structure of a hecke ring of a chevalley group over a finite field. J. Fac. Sci. Univ. Tokyo, 10:215–236, 1964.MATHMathSciNetGoogle Scholar
  21. 75.
    S. Janson. Poisson approximation for large deviations. Random Struct. Algorithms, 1:221–229, 1990.MATHCrossRefMathSciNetGoogle Scholar
  22. 80.
    E.Y. Jin, C.M. Reidys, and R.R. Wang. Asymptotic analysis of k-noncrossing matchings. arXiv:0803.0848, 2008.Google Scholar
  23. 83.
    V.F. Kolchin. Random Mappings. Number 14 in Translations Series. Optimization Software, New York, 1986.Google Scholar
  24. 98.
    A.M. Odlyzko. Handbook of combinatorics. Elsevier, Amsterdam, 2005.Google Scholar
  25. 102.
    C.M. Reidys. Random induced subgraphs of generalized n-cubes. Adv. Appl. Math., 19:360–377, 1997.MATHCrossRefMathSciNetGoogle Scholar
  26. 105.
    C.M. Reidys. The largest component in random induced subgraphs of n-cubes. Discr. Math., 309, Issue 10:3113–3124, 2009.MATHCrossRefMathSciNetGoogle Scholar
  27. 106.
    C.M. Reidys, P.F. Stadler, and P.K. Schuster. Generic properties of combinatory maps and neutral networks of rna secondary structures. Bull. Math. Biol., 59(2):339–397, 1997.MATHCrossRefGoogle Scholar
  28. 115.
    C.E. Schensted. Longest increasing and decreasing subsequences. Canad. J. Math., 13:179–191, 1961.MATHMathSciNetGoogle Scholar
  29. 125.
    R.P. Stanley. Differentiably finite power series. Eur. J. Combinator., 1:175–188, 1980.MATHMathSciNetGoogle Scholar
  30. 127.
    R.P. Stanley. Enumerative Combinatorics, volume 2. Cambridge University Press, Cambridge, England, 2000.Google Scholar
  31. 129.
    S. Sundaram. The Cauchy identity for Sp(2n). J. Comb. Theory, Ser. A, 53:209–238, 1990.MATHCrossRefMathSciNetGoogle Scholar
  32. 140.
    W. Wasow. Asymptotic Expansions for Ordinary Differential Equations. Dover, 1987.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Research Center for CombinatoricsNankai UniversityTianjinChina

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