Parenthesization/Examples/4

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Example of Parenthesization

A word of $4$ elements can be parenthesized in $5$ distinct ways:

$\quad a_1 \paren {a_2 \paren {a_3 a_4} }$
$\quad a_1 \paren {\paren {a_2 a_3} a_4}$
$\quad \paren {a_1 a_2} \paren {a_3 a_4}$
$\quad \paren {a_1 \paren {a_2 a_3} } a_4$
$\quad \paren {\paren {a_1 a_2} a_3} a_4$

Proof

From Number of Distinct Parenthesizations on Word, the number of distinct parenthesizations of a word $w$ of $n$ elements is the Catalan number $C_{n - 1}$:

$C_{n - 1} = \dfrac 1 n \dbinom {2 \paren {n - 1} } {n - 1}$

For $n = 4$ we have:

 $\ds C_4$ $=$ $\ds \dfrac 1 4 \dbinom {2 \times 3} 3$ $\ds$ $=$ $\ds \dfrac 1 4 \times \dfrac {6!} {3! \times 3!}$ Definition of Binomial Coefficient $\ds$ $=$ $\ds \dfrac 1 4 \times \dfrac {6 \times 5 \times 4} {3 \times 2 \times 1}$ Definition of Factorial $\ds$ $=$ $\ds 5$

$\blacksquare$