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.
“Like the effects of industrial pollution ... the AIDS crisis is evidence of a world in which nothing important is regional, local, limited; in which everything that can circulate does, and every problem is, or is destined to become, worldwide.”
—Susan Sontag (b. 1933)