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:
“The table kills more people than war does.”
—Catalan proverb, quoted in Colman Andrews, Catalan Cuisine.
“My problem lies in reconciling my gross habits with my net income.”
—Errol Flynn (1909–1959)