Catalan Numbers

  • Peter Hilton
  • Derek Holton
  • Jean Pedersen
Part of the Undergraduate Texts in Mathematics book series (UTM)


The classical Catalan numbers, rediscovered by the Belgian mathematician Eugène Charles Catalan (1814–1894) seem to have been first studied by the famous Swiss mathematician Leonhard Euler (1707–1783). Euler considered the problem of counting the number of ways a given convex polygon1 can be divided into triangles by drawing non-intersecting diagonals (a diagonal is a line segment joining non-adjacent vertices, and two diagonals are considered to be non-intersecting if they intersect only in a vertex of the polygon). Obviously, this number depends only on n, the number of sides of the polygon; obviously, too, (n - 3) diagonals will be drawn, creating (n - 2) triangles. We will call the number2 c n-2, thus putting emphasis on the number of triangles; this accords with standard practice. It is easy to see that, for polygons with 3,4, 5 sides, we have d = 1, c 2 = 2, c 3 = 5, respectively (see Figure 1). Even Euler, however, found it difficult to obtain a general formula for c k .


Source Node Binary Tree Good Path Catalan Number Binomial Theorem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    André, D, Solution directe du problème résolu par M. Bertrand, Comptes Rendus Acad. Sci. Paris 105 (1887), 436–7.Google Scholar
  2. 2.
    Gould, Henry W., Bell and Catalan Numbers, Research bibliographies 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.Google Scholar
  3. 3.
    Hilton, Peter, Derek Holton, and Jean Pedersen, Mathematical Reflections — In a Room with Many Mirrors, 2nd printing, Springer-Verlag, NY, 1998.Google Scholar
  4. 4.
    Hilton, Peter, and Jean Pedersen, Extending the binomial coefficients to preserve symmetry and pattern, Computers Math. Applic. 17 (1–3) (1989), 89–102.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Hilton, Peter, and Jean Pedersen, Catalan numbers, their generalizations and their uses, The Mathematical Intelligencer 13, No. 2 (1991), 64–75.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Hilton, Peter, Jean Pedersen, and Tamsen Whitehead, On paths on the integral lattice in the plane, Far East Journal of Mathematical Sciences, Feb. (1999), 1–23.Google Scholar
  7. 7.
    Larcombe, P. J., and P.D.C. Wilson, On the trail of the Catalan sequence, Mathematics Today, 35, No. 1 (1998), 114–117.Google Scholar
  8. 8.
    Sloane, N.J.A., and Simon Plouffe, The Encyclopedia of Integer Sequences, New York and London, Academic Press, 1995.MATHGoogle Scholar
  9. 9.
    Stanley, Richard P., Enumerative Combinatorics, vol. 2, Cambridge University Press, 1999.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Peter Hilton
    • 1
  • Derek Holton
    • 2
  • Jean Pedersen
    • 3
  1. 1.Mathematical Sciences DepartmentSUNY at BinghamtonBinghamtonUSA
  2. 2.Department of Mathematics and StatisticsUniversity of OtagoDunedinNew Zealand
  3. 3.Department of Mathematics and Computer ScienceSanta Clara UniversitySanta ClaraUSA

Personalised recommendations