Take itself to be a bracketing, then recursively define a bracketing as a sequence
where and
each is a bracketing. A bracketing can be represented as a parenthesized string of s, with parentheses removed from
any single letter for clarity of notation (Stanley 1997). Bracketings built up of binary operations only are called
Binary Bracketings. For example, four letters have 11 possible bracketings:

the last five of which are binary.

The number of bracketings on letters is given by the Generating Function

(Schröder 1870, Stanley 1997) and the Recurrence Relation

(Sloane), giving the sequence for as 1, 1, 3, 11, 45, 197, 903, ... (Sloane's A001003). The numbers are also given by

for (Stanley 1997).

The first Plutarch Number 103,049 is equal to (Stanley 1997), suggesting that
Plutarch's problem of ten compound propositions is equivalent to the number of bracketings. In addition, Plutarch's second
number 310,954 is given by
(Habsieger *et al. *1998).

**References**

Habsieger, L.; Kazarian, M.; and Lando, S. ``On the Second Number of Plutarch.'' *Amer. Math. Monthly* **105**, 446, 1998.

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

Sloane, N. J. A. Sequence
A001003/M2898
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and 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