Parenthesization/Examples/4

Example of Parenthesization
A word of $4$ elements can be parenthesized in $5$ distinct ways:
 * $\quad a_1 \left({a_2 \left({a_3 a_4}\right)}\right)$
 * $\quad a_1 \left({\left({a_2 a_3}\right) a_4}\right)$
 * $\quad \left({a_1 a_2}\right) \left({a_3 a_4}\right)$
 * $\quad \left({a_1 \left({a_2 a_3}\right)}\right) a_4$
 * $\quad \left({\left({a_1 a_2}\right) a_3}\right) a_4$

Proof
From Number of Distinct Parenthesizations on Word, the number of distinct parenthesizations of a word $w$ of $n$ elements is the Catalan number $C_{n - 1}$:
 * $C_{n - 1} = \dfrac 1 n \dbinom {2 \left({n - 1}\right)} {n - 1}$

For $n = 4$ we have: