The Catalan numbers have many interpretations in combinatorics, two of them being the number of ways to insert parentheses into a sequence of numbers so that they are grouped into pairs—for example, ((12)3) and (1(23))—and the number of ways to divide a polygon into triangles.

The Fuss–Catalan numbers are a generalization of the Catalan numbers and have many analogous interpretations. For example, two interpretations of the Fuss–Catalan numbers are the number of ways to insert parentheses in a sequence of numbers so that they are grouped into triples—for example, ((123)45), (1(234)5) and (12(345))—and the number of ways to divide a polygon with an even number of sides into quadrilaterals. This Demonstration shows the equivalence of these two Fuss–Catalan interpretations by labeling each polygon division with the corresponding parenthesized expression.

[1] R. P. Stanley, Enumerative Combinatorics, Vol. 2, New York: Cambridge University Press, 1999 pp. 212, 234 (Exercise 6.33(c)).

[2] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed., Reading, MA: Addison-Wesley, 1994 p. 361.

[3] M. Gardner, Time Travel and Other Mathematical Bewilderments, New York: W. H. Freeman, 1988 Chapter 20.