In mathematics, Catalan's problem asks the number of ways n + 1 factors can be completely parenthesized by n pairs of parentheses. For example, the following are the 14 ways that 5 factors can be parenthesized:
- (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)
The numbers of ways of performing these pairings are the Catalan numbers.
Famous quotes containing the words catalan and/or problem:
“It’s better that it should make you sick than that you don’t eat it at all.”
—Catalan proverb, quoted in Colman Andrews, Catalan Cuisine.
“Give a scientist a problem and he will probably provide a solution; historians and sociologists, by contrast, can offer only opinions. Ask a dozen chemists the composition of an organic compound such as methane, and within a short time all twelve will have come up with the same solution of CH4. Ask, however, a dozen economists or sociologists to provide policies to reduce unemployment or the level of crime and twelve widely differing opinions are likely to be offered.”
—Derek Gjertsen, British scientist, author. Science and Philosophy: Past and Present, ch. 3, Penguin (1989)