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.
“Great speeches have always had great soundbites. The problem now is that the young technicians who put together speeches are paying attention only to the soundbite, not to the text as a whole, not realizing that all great soundbites happen by accident, which is to say, all great soundbites are yielded up inevitably, as part of the natural expression of the text. They are part of the tapestry, they aren’t a little flower somebody sewed on.”
—Peggy Noonan (b. 1950)