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 first Plutarch number 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
is given by
(Habsieger et al. 1998).

Comtet, L. "Bracketing Problems." §1.15 in Advanced
Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht,
Netherlands: Reidel, pp. 52-57, 1974.Habsieger, L.; Kazarian, M.;
and Lando, S. "On the Second Number of Plutarch." Amer. Math. Monthly105,
446, 1998.Schröder, E. "Vier combinatorische Probleme."
Z. Math. Physik15, 361-376, 1870.Sloane, N. J. A.
Sequence A001003/M2898 in "The On-Line
Encyclopedia of Integer Sequences."Stanley, R. P. "Hipparchus,
Plutarch, Schröder, and Hough." Amer. Math. Monthly104,
344-350, 1997.Vardi, I. Computational
Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 198-199,
1991.