Binary Bracketing
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 Catalanin 1838. For example, for the four letters 
, 
, 
, and 
there are five possibilities: 
, 
,
, 
, and 
, written in shorthand as 
, 
, 
, 
, and 
.See also Bracketing, Catalan Number, Catalan's Problem
References
Schröder, E. ``Vier combinatorische Probleme.'' Z. Math. Physik 15, 361-376, 1870. Sloane, N. J. A. SequenceA000108/M1459in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and extended entry inSloane, 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.
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