A nonassociative product uses a binary operation that is not associative. Its possible binary bracketings specify the order of evaluation. Different bracketings need not give distinct values.
For integer 
, let 
count binary bracketings
of 
ordered elements with exactly 
elements preceding the rightmost left parenthesis.
For integer 
with 
,
 |
(1)
|
where 
is a binomial coefficient and 
. In particular, 
and 
. These counts satisfy the recurrence
equation
 |
(2)
|
for 
and 
(Niven 1965, pp. 146-150).
The number of binary bracketings of an 
-fold
product in a nonassociative algebra is
 |
(3)
|
for 
,
with 
,
where 
is a Catalan number. For 
, 2, ..., the counts are 1, 1, 2, 5, 14, 42, 132, ... (OEIS
A000108).
