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GeezerVison is showing mathworld.wolfram.com — Catalan's Problem -- from Wolfram MathWorld

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Catalan's Problem

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The problem of finding the number of different ways in which a product of n different ordered factors can be calculated by pairs (i.e., the number of binary bracketings of n letters). For example, for the four factors a, b, c, and d, there are five possibilities: ((ab)c)d, (a(bc))d, (ab)(cd), a((bc)d), and a(b(cd)).

The solution was given by Catalan in 1838 as

C_n^'=((4n-6)!!!!)/(n!)
(1)
=(2·6·10...(4n-6))/(n!)
(2)
=C_(n-1),
(3)

where n!!!! is a multifactorial, n! is the usual factorial, and C_n is a so-called Catalan number.

See also

Binary Bracketing, Catalan Number, Catalan's Diophantine Problem, Euler's Polygon Division Problem

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References

Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 23, 1965.

Referenced on Wolfram|Alpha

Catalan's Problem

Cite this as:

Weisstein, Eric W. "Catalan's Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CatalansProblem.html

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