Catalan Numbers

The Catalan numbers (1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, ...), named after Eugène Charles Catalan (1814–1894), arise in a number of problems in combinatorics. They can be computed using this formula:

C(2n, n)/(n + 1)

Among other things, the Catalan numbers describe:

The following figures show some of the interpretations—polygon divisions, binary trees, pairwise multiplication—combined. For n=2:

interpretations as square division, tree, multiplication

And n=3:

interpretations as pentagon division, tree, multiplication

Polygon diagrams:

4 sides, 2 ways:

two square triangulations: SW-to-NE and NW-to-SE

5 sides, 5 ways:

pentagon triangulations

6 sides, 14 ways:

hexagon triangulations

7 sides, 42 ways:

septagon triangulations

8 sides, 132 ways:

octagon triangulations

9 sides, 429 ways:
(Hidden in file catalan9.png, as we used to do.)

Step diagrams:

2 rectangles, 2 ways:

3 rectangles, 5 ways:

4 rectangles, 14 ways:

5 rectangles, 42 ways:

6 rectangles, 132 ways:

Multiplication diagrams:

3 numbers:

(1 (2 3))   ((1 2) 3)

4 numbers:

(1 (2 (3 4)))   (1 ((2 3) 4))
((1 2) (3 4))   ((1 (2 3)) 4)
(((1 2) 3) 4)

5 numbers:

(1 (2 (3 (4 5))))   (1 (2 ((3 4) 5)))
(1 ((2 3) (4 5)))   (1 ((2 (3 4)) 5))
(1 (((2 3) 4) 5))   ((1 2) (3 (4 5)))
((1 2) ((3 4) 5))   ((1 (2 3)) (4 5))
((1 (2 (3 4))) 5)   ((1 ((2 3) 4)) 5)
(((1 2) 3) (4 5))   (((1 2) (3 4)) 5)
(((1 (2 3)) 4) 5)   ((((1 2) 3) 4) 5)

6 numbers:

(1 (2 (3 (4 (5 6)))))   (1 (2 (3 ((4 5) 6))))
(1 (2 ((3 4) (5 6))))   (1 (2 ((3 (4 5)) 6)))
(1 (2 (((3 4) 5) 6)))   (1 ((2 3) (4 (5 6))))
(1 ((2 3) ((4 5) 6)))   (1 ((2 (3 4)) (5 6)))
(1 ((2 (3 (4 5))) 6))   (1 ((2 ((3 4) 5)) 6))
(1 (((2 3) 4) (5 6)))   (1 (((2 3) (4 5)) 6))
(1 (((2 (3 4)) 5) 6))   (1 ((((2 3) 4) 5) 6))
((1 2) (3 (4 (5 6))))   ((1 2) (3 ((4 5) 6)))
((1 2) ((3 4) (5 6)))   ((1 2) ((3 (4 5)) 6))
((1 2) (((3 4) 5) 6))   ((1 (2 3)) (4 (5 6)))
((1 (2 3)) ((4 5) 6))   ((1 (2 (3 4))) (5 6))
((1 (2 (3 (4 5)))) 6)   ((1 (2 ((3 4) 5))) 6)
((1 ((2 3) 4)) (5 6))   ((1 ((2 3) (4 5))) 6)
((1 ((2 (3 4)) 5)) 6)   ((1 (((2 3) 4) 5)) 6)
(((1 2) 3) (4 (5 6)))   (((1 2) 3) ((4 5) 6))
(((1 2) (3 4)) (5 6))   (((1 2) (3 (4 5))) 6)
(((1 2) ((3 4) 5)) 6)   (((1 (2 3)) 4) (5 6))
(((1 (2 3)) (4 5)) 6)   (((1 (2 (3 4))) 5) 6)
(((1 ((2 3) 4)) 5) 6)   ((((1 2) 3) 4) (5 6))
((((1 2) 3) (4 5)) 6)   ((((1 2) (3 4)) 5) 6)
((((1 (2 3)) 4) 5) 6)   (((((1 2) 3) 4) 5) 6)

Tree diagrams:

3 nodes:

[ again, not much without pictures -- sorry ]

4 nodes:

5 nodes:

6 nodes:

Path diagrams:

2 × 2 grid:

[ Again, a picture speaks a thousand etc. ]

3 × 3 grid:

4 × 4 grid:

5 × 5 grid:

The Catalan interpretations can be partially ordered and arranged into a lattice called a Tamari lattice:

Tamari lattice of triangulations

You might also enjoy the generalization Fuss–Catalan numbers.

Originally designed and rendered using Mathematica 3.0 for the Apple Macintosh.

Inspiration and facts (though not figures) by Brian Hayes, A Question of Numbers [dead link], American Scientist, January–February 1996; Steven S. Skiena, Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica, Addison-Wesley, 1990; Fred S. Roberts, Applied Combinatorics, Prentice-Hall, 1984; D. E. Knuth, Sorting and Searching (vol. 3 of The Art of Computer Programming), Addison-Wesley, 1973. Catalan dates from Florian Cajori, A History of Mathematics, The Macmillan Company, 1922; R. P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge: Cambridge University Press, 1999.

See also Martin Gardner, Time Travel and Other Mathematical Bewilderments, Chapter 20, W. H. Freeman, 1988; and Ilan Vardi, Computational Recreations in Mathematica, Chapter 9, Addison-Wesley, 1991.

Copyright © 1996–2017 by Robert Dickau.

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