Skip to main content
Log in

Catalan Numbers, Their Generalization, and Their Uses

  • Articles
  • Published:
The Mathematical Intelligencer Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. D. André, Solution directe du problème résolu par M. Bertrand,Comptes Rendus Acad. Sci. Paris 105 (1887), 436–7.

    Google Scholar 

  2. A. P. Hillman, G. L. Alexanderson and R. M. Grassl,Discrete and Combinatorial Mathematics, San Francisco: Dellen (1987).

    MATH  Google Scholar 

  3. Peter Hilton and Jean Pedersen, Extending the binomial coefficients to preserve symmetry and pattern,Computers Math. Applic. 17 (1–3) (1989), 89–102.

    Article  MATH  MathSciNet  Google Scholar 

  4. Peter Hilton and Jean Pedersen, The ballot problem and Catalan numbers,Nieuw Archief voor Wiskunde, 1990 (to appear).

  5. A. Hurwitz and R. Courant,Allgemeine Funktionentheorie und elliptische Funktionen. Geometrische Funktionentheorie, Berlin: Springer (1922), 128.

    Book  Google Scholar 

  6. David A. Klarner, Correspondences between plane trees and binary sequences,J. of Comb. Theory 9 (1970), 401–411.

    Article  MATH  MathSciNet  Google Scholar 

  7. G. Polya and G. Szegö,Aufgaben und Lehrsätze aus der Analysis, Vol. I, Berlin, Göttingen, Heidelberg: Springer (1954), 125.

    Book  Google Scholar 

  8. N. J. A. Sloane,A Handbook of Integer Sequences, New York and London: Academic Press (1973).

    MATH  Google Scholar 

  9. Marta Sved, Counting and recounting,Mathematical Intelligencer 5, no. 4 (1983), 21–26.

    Article  MATH  MathSciNet  Google Scholar 

Bibliography [Items [2], [6] and [8] among the References may also be regarded as general sources on Catalan numbers].

  • R. Alter, Some remarks and results on Catalan numbers,Proc. 2nd Louisiana Conf. on Comb., Graph Theory and Comp. (1971), 109–132.

  • D. André, Solution directe du problème résolu par M. Bertrand,Comptes Rendus Acad. Sci. Paris 105 (1887), 436–7.

    Google Scholar 

  • W. G. Brown, Historical note on a recurrent combinatorial problem,Amer. Math. Monthly 72 (1965), 973–977.

    Article  MATH  MathSciNet  Google Scholar 

  • Wenchang Chu, A new combinatorial interpretation for generalized Catalan numbers,Discrete Math. 65 (1987), 91–94.

    Article  MATH  MathSciNet  Google Scholar 

  • K. L. Chung and W. Feller, On fluctuations in coin-tossing,Proc. Nat. Acad. Sci., U.S.A. 35 (1949), 605–608.

    Article  MATH  MathSciNet  Google Scholar 

  • John H. Conway and Richard K. Guy,The Book of Numbers, Sci. Amer. Library, W. H. Freeman (to appear 1990).

  • I. Z. Chorneyko and S. G. Mohanty, On the enumeration of certain sets of planted trees,J. Combin. Theory Ser. B18 (1975), 209–21.

    Article  MathSciNet  Google Scholar 

  • T. T. Cong and M. Sato, One-dimensional random walk with unequal step lengths restricted by an absorbing barrier,Discrete Math. 40 (1982), 153–162.

    Article  MATH  MathSciNet  Google Scholar 

  • R. Donoghey and L. W. Shapiro, Motzkin numbers,J. Combin. Theory Ser. A23 (1977), 291–301.

    Article  Google Scholar 

  • N. Dershowitz and S. Zaks, Enumeration of ordered trees,Discrete Math. 31 (1980), 9–28.

    Article  MATH  MathSciNet  Google Scholar 

  • L. Euler,Novi Commentarii Academiae Scientarium Imperialis Petropolitanque 7 (1758/9), 13–14.

    Google Scholar 

  • Roger B. Eggleton and Richard K. Guy, Catalan strikes again! How likely is a function to be convex?,Math. Mag. 61 (1988), 211–219.

    Article  MATH  MathSciNet  Google Scholar 

  • A. Erdélyi and I. M. H. Etherington, Some problems of non-associative combinatorics,Edinburgh Math. Notes. 32 (1941), 7–12.

    MATH  Google Scholar 

  • I. J. Good, Generalizations to several variables of Lagrange’s expansion,Proc. Camb. Phil. Soc. 56 (1960), 367–380.

    Article  MATH  MathSciNet  Google Scholar 

  • Henry W. Gould,Bell and Catalan Numbers, Research bibliography of two special number sequences, available from the author (Department of Mathematics, West Virginia University, Morgantown, WV 26506). The 1979 edition sells for $3.00 and contains over 500 references pertaining to Catalan numbers.

  • Henry W. Gould,Combinatorial Identities, available from the author (Department of Mathematics, West Virginia University, Morgantown, WV 26506).

  • Richard K. Guy, Dissecting a polygon into triangles,Bulletin of the Malayan Mathematical Society 5 (45) (1958), 57–60.

    Google Scholar 

  • Richard K. Guy, A medley of Malayan mathematical memories and mantissae,Math. Medley (Singapore), 12, no. 1 (1984), 9–17.

    Google Scholar 

  • V. E. Hoggatt, Jr., and Marjorie Bicknell, Pascal, Catalan, and general sequence convolution arrays in a matrix,Fibonacci Quarterly 14, no. 2, 135–143.

  • V. E. Hoggatt, Jr., and Marjorie Bicknell, Sequences of matrix inverses from Pascal, Catalan, and related convolution arrays,Fibonacci Quarterly 14, no. 3, 224–232.

  • V. E. Hoggatt, Jr., and Marjorie Bicknell, Catalan and related sequences arising from inverses of Pascal’s triangle matrices,Fibonacci Quarterly (December 1976), 395–405.

  • M. S. Klamkin, Problem 4983,Amer. Math. Monthly 69 (1962), 930–931.

    Article  MathSciNet  Google Scholar 

  • Mike Kuchinski,Catalan structures and Correspondences, M.Sc. thesis, Department of Mathematics, West Virginia University, Morgantown, WV 26506.

  • Th. Motzkin, Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, and for non-associative products,Bull. Amer. Math. Soc. 54 (1948), 352–360.

    Article  MATH  MathSciNet  Google Scholar 

  • G. Polya, On picture-writing,Amer. Math. Monthly 63 (1956), 689–697.

    Article  MATH  MathSciNet  Google Scholar 

  • G. N. Raney, Functional composition patterns and power series inversion,Trans. Amer. Math. Soc. 94 (1960), 441–451.

    Article  MATH  MathSciNet  Google Scholar 

  • D. G. Rogers, Pascal triangles, Catalan numbers and renewal arrays,Discrete Math. 22 (1978), 301–310.

    Article  MATH  MathSciNet  Google Scholar 

  • A. D. Sands, On generalized Catalan numbers,Discrete Math. 21 (1978), 218–221.

    Article  MathSciNet  Google Scholar 

  • Memoirs by Eugène-Charles Catalan relevant to the theme of Catalan numbers may be found inJournal de Mathématiques pures et appliquées de Liouville, (1), III, 508-816; IV, 91-94, 95-99; VI, 74, 1838-41.

  • For details of Catalan’s life and work see “Les travaux mathématiques de Eugène-Charles Catalan,”Annuaire de L’Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique, Brussels (1896), 115–172.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hilton, P., Pedersen, J. Catalan Numbers, Their Generalization, and Their Uses. The Mathematical Intelligencer 13, 64–75 (1991). https://doi.org/10.1007/BF03024089

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03024089

Keywords

Navigation