Skip to main content

Enumeration and Special Functions

  • 1188 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1817)

Abstract

These notes for the August 12–16, 2002 Euro Summer School in OPSF at Leuven have three sections with basic introductions to

  1. 1

    Enumeration and q -series

  2. 2

    Enumeration and orthogonal polynomials

  3. 3

    Symmetric functions. No prior exposure to these areas is assumed. Three excellent textbooks for these three topics are [1], [7], and [17]. Several exercises and open problems are given throughout these notes.

Keywords

  • Orthogonal Polynomial
  • Lattice Path
  • Plane Partition
  • Combinatorial Interpretation
  • Ferrers Diagram

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Research partially supported by NSF grant DMS 0203282

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/3-540-44945-0_4
  • Chapter length: 30 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   44.99
Price excludes VAT (USA)
  • ISBN: 978-3-540-44945-4
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   59.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and Its Applications, Vol. 2 (G.-C. Rota ed.), Addison-Wesley, Reading, Mass., 1976 (reissued by Cambridge Univ. Press, London and New York, 1985).

    Google Scholar 

  2. ___, On a conjecture of Peter Borwein, J. Symbolic Computation 20 (1995), 487–501.

    MATH  CrossRef  Google Scholar 

  3. ___, On the proofs of the Rogers-Ramanujan identities, in ‘q-Series and Partitions’, IMA Vol. Math. 18, Springer, New York, 1989, pp. 1–14.

    Google Scholar 

  4. G. Andrews, R. Baxter, D. Bressoud, W. Burge, P. Forrester, G. Viennot, Partitions with prescribed hook differences, Eur. J. Comb. 8 (1987), 341–350.

    MATH  MathSciNet  Google Scholar 

  5. D. Bressoud, The Borwein conjecture and partitions with prescribed hook differences, Elec. J. Comb. 3 (1996), 14 pp.

    MathSciNet  Google Scholar 

  6. ___, Proofs and confirmations. The story of the alternating sign matrix conjecture, Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  7. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.

    MATH  Google Scholar 

  8. P. Flajolet, Combinatorial aspects of continued fractions, Disc.Math. 32 (1980), 125–161.

    MATH  MathSciNet  Google Scholar 

  9. D. Foata, A combinatorial proof of the Mehler formula, J. Comb. A 24 (1978), 367–376.

    MATH  CrossRef  MathSciNet  Google Scholar 

  10. D. Foata and P. Leroux, Polynômes de Jacobi, interpretation combinatoire et fonction génératrice, Proc. Amer. Math. Soc. 87 (1983), 47–53.

    Google Scholar 

  11. D. Foata and V. Strehl, Combinatorics of Laguerre polynomials, in ‘Enumeration and Design’, Academic Press, Toronto, 1984, pp. 123–140.

    Google Scholar 

  12. A. Garsia and S. Milne, Method for constructing bijections for classical partition identities, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), 2026–2028.

    Google Scholar 

  13. F. Garvan, D. Kim, and D. Stanton, Cranks and t-cores, Inv.Math. 101 (1990), 1–17.

    MATH  CrossRef  MathSciNet  Google Scholar 

  14. G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.

    MATH  Google Scholar 

  15. I. Gessel and G. Viennot, Binomial determinants, paths, and hook length formulae, Adv. in Math. 58 (1985), 300–321.

    CrossRef  MathSciNet  Google Scholar 

  16. M. Ismail, D. Stanton, and G. Viennot, The combinatorics of the q-Hermite polynomials and the Askey-Wilson integral, Eur. J. Comb. 8 (1987), 379–392.

    MATH  MathSciNet  Google Scholar 

  17. I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, Oxford, 1995.

    MATH  Google Scholar 

  18. ___, Affine root systems and Dedekind’s η-function, Inv. Math. 15 (1972), 91–143.

    Google Scholar 

  19. K. O’Hara, Unimodality of Gaussian coefficients: a constructive proof, J. Comb. A 53 (1990), 29–52.

    MATH  CrossRef  MathSciNet  Google Scholar 

  20. I. Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche, reprinted in ‘I. Schur, Gesammelte Abhandlungen’, volume 2, Springer, Berlin, 1973, pp. 117–136.

    Google Scholar 

  21. R. Simion and D. Stanton, Octabasic Laguerre polynomials and permutation statistics, J. Comput. Appl. Math. 68 (1996), 297–329.

    MATH  CrossRef  MathSciNet  Google Scholar 

  22. R. Stanley, Enumerative Combinatorics, Wadsworth, Monterey, 1986.

    Google Scholar 

  23. ___, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Alg. Disc. Methods 1 (1980), 168–183.

    Google Scholar 

  24. D. Stanton, Gaussian integrals and the Rogers-Ramanujan identities, in ‘Symbolic computation, number theory, special functions, physics, and combinatorics’ (F. Garvan and M. Ismail, eds.), Kluwer, Dordrecht, 2001, pp. 255–266.

    Google Scholar 

  25. G. Viennot, Une Théorie Combinatoire des Polynômes Orthogonaux Généraux, Lecture Notes, University of Quebec at Montreal, 1983.

    Google Scholar 

  26. D. Zeilberger, A one-line high school algebra proof of the unimodality of the Gaussian polynomials [n k]for k 6lt; 20, in ‘q-Series and Partitions’, IMA Vol. Math. 18, Springer, New York, 1989, pp. 67–72.

    Google Scholar 

  27. ___, A q-Foata proof of the q-Saalschütz identity, Eur. J. Comb. 8 (1987), 461–463.

    MATH  MathSciNet  Google Scholar 

  28. ___, Proof of the alternating sign matrix conjecture, Elec. J. Comb. 3 (1996), 1–84.

    Google Scholar 

  29. D. Zeilberger and D. Bressoud, A proof of Andrews’ q-Dyson conjecture, Disc. Math. 54 (1985), 201–224.

    MATH  CrossRef  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Stanton, D. (2003). Enumeration and Special Functions. In: Koelink, E., Van Assche, W. (eds) Orthogonal Polynomials and Special Functions. Lecture Notes in Mathematics, vol 1817. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44945-0_4

Download citation

  • DOI: https://doi.org/10.1007/3-540-44945-0_4

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40375-3

  • Online ISBN: 978-3-540-44945-4

  • eBook Packages: Springer Book Archive