## 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 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.