A binary bracketing is a Bracketing built up entirely of binary operations. The number of binary bracketings of letters
(Catalan's Problem) are given by the Catalan Numbers , where

where denotes a Binomial Coefficient and is the usual Factorial, as first shown by Catalan in 1838. For example, for the four letters , , , and there are five possibilities: , , , , and , written in shorthand as , , , , and .

**References**

Schröder, E. ``Vier combinatorische Probleme.'' *Z. Math. Physik* **15**, 361-376, 1870.

Sloane, N. J. A. Sequence
A000108/M1459
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and extended entry in
Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Stanley, R. P. ``Hipparchus, Plutarch, Schröder, and Hough.'' *Amer. Math. Monthly* **104**, 344-350, 1997.

© 1996-9

1999-05-26