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 .
Schröder, E. "Vier combinatorische Probleme." Z. Math. Physik15, 361-376, 1870.Sloane, N. J. A.
Sequence A000108/M1459 in "The On-Line
Encyclopedia of Integer Sequences."Sloane, N. J. A. and
Plouffe, S. Figure M1459 in The
Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.Stanley,
R. P. "Hipparchus, Plutarch, Schröder, and Hough." Amer. Math.
Monthly104, 344-350, 1997.
letters (Catalan's problem) are given by the
Catalan numbers 
, 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 
.
