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Probabilistic Analysis

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Combinatorial Computational Biology of RNA

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Abstract

In this section we prove that k-noncrossing RNA structures can be generated efficiently with uniform probability. The results presented here are derived from [26] and are based on Section 2.1.

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References

  1. E.A. Bender. Central and local limit theorem applied to asymptotic enumeration. J. Comb. Theory, Ser. A, 15:91–111, 1973.

    Article  MATH  MathSciNet  Google Scholar 

  2. N.T. Cameron and L. Shapiro. Random walks, trees and extensions of Riordan group techniques. Talk, in: Annual Joint Mathematics Meetings, Baltimore, MD, 2003.

    Google Scholar 

  3. 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.

    Article  MATH  MathSciNet  Google Scholar 

  4. W.Y.C. Chen, H.S.W. Han, and C.M. Reidys. Random k-noncrossing RNA structures. Proc. Natl. Acad. Sci. USA, 106(52):22061–22066, 2009.

    Article  MathSciNet  Google Scholar 

  5. E. Deutsch and L. Shapiro. A survey of the fine numbers. Discrete Math., 241:241–265, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  6. P. Duchon, P. Flajolet, G. Louchard, and G. Schaeffer. Boltzmann samplers for the random generation of combinatorial structures. Combin. Probab. Comput., 13:577–625, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  7. W. Feller. An introduction to probability theory and its application. Addison-Wesley Publishing Company Inc., NY, 1991.

    Google Scholar 

  8. P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge University Press, Cambridge, England, 2009.

    MATH  Google Scholar 

  9. I.M. Gessel and X.G. Viennot. Determinants, paths, and plane partitions. preprint, 1989.

    Google Scholar 

  10. D. Gouyou-Beauschamps. Standard young tableaux of height 4 and 5. Europ. J. Combin., 10:69–82, 1989.

    Google Scholar 

  11. H.S.W. Han and C.M. Reidys. Stacks in canonical RNA pseudoknot structures. Math. Bioscience, 219, Issue 1:7–14, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  12. I.L. Hofacker. Vienna RNA secondary structure server. Nucl. Acids. Res., 31(13):3429–3431, 2003.

    Article  Google Scholar 

  13. F.W.D. Huang and C.M. Reidys. Statistics of canonical RNA pseudoknot structures. J. Theor. Biol., 253:570–578, 2008.

    Article  Google Scholar 

  14. E.Y. Jin and C.M. Reidys. Asymptotic enumeration of RNA structures with pseudoknots. Bull. Math. Biol., 70:951–970, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  15. E.Y. Jin and C.M. Reidys. RNA pseudoknots structures with arc length-length \(\ge 3\) and stack-length-length \(\ge \sigma\). Discr. Appl. Math., 158:25–36, 2010.

    Article  MATH  MathSciNet  Google Scholar 

  16. R.B. Lyngsø and C.N.S. Pedersen. RNA pseudoknot prediction in energy-based models. J. Comput. Biol., 7:409–427, 2000.

    Article  Google Scholar 

  17. M. Renault. Lost (and found) in translation: André’s actual method and its application to the generalized ballot problem. Amer. Math. Monthly., 115:358–363, 2008.

    MATH  MathSciNet  Google Scholar 

  18. B. Salvy and P. Zimmerman. Gfun: a maple package for the manipulation of generating and holonomic functions in one variable. ACM TOMS, 20:163–177, 1994.

    Article  MATH  Google Scholar 

  19. L. Shapiro, S. Getu, W. Woan, and L. Woodson. The Riordan group. Discr. Appl. Math., 34:229–239, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  20. R.P. Stanley. Differentiably finite power series. Eur. J. Combinator., 1:175–188, 1980.

    MATH  MathSciNet  Google Scholar 

  21. H.S. Wilf. A unified setting for sequencing, ranking, and selection algorithms for combinatorial objects. Adv. Math., 24:281–291, 1977.

    MATH  MathSciNet  Google Scholar 

  22. H.S. Wilf. Combinatorial algorithms. Academic Press, NY, 1978.

    MATH  Google Scholar 

  23. D. Zeilberger. A holonomic systems approach to special functions identities. J. Comput. Appl. Math., 32:321–368, 1990.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Research Center for Combinatorics, Nankai University, Tianjin, 300071, China

    Christian Reidys

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  1. Christian Reidys
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Correspondence to Christian Reidys .

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Reidys, C. (2011). Probabilistic Analysis. In: Combinatorial Computational Biology of RNA. Springer, New York, NY. https://doi.org/10.1007/978-0-387-76731-4_5

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