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.
“The great problem of American life [is] the riddle of authority: the difficulty of finding a way, within a liberal and individualistic social order, of living in harmonious and consecrated submission to something larger than oneself.... A yearning for self-transcendence and submission to authority [is] as deeply rooted as the lure of individual liberation.”
—Wilfred M. McClay, educator, author. The Masterless: Self and Society in Modern America, p. 4, University of North Carolina Press (1994)