Parenthesization/Examples/3

Example of Parenthesization

A word of $3$ elements can be parenthesized in $2$ distinct ways:

$\quad a_1 \left({a_2 a_3}\right)$

$\quad \left({a_1 a_2}\right) a_3$

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 $n = 3$ we have:

\(\ds C_2\)

\(=\)

\(\ds \dfrac 1 3 \dbinom {2 \times 2} 2\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 3 \times \dfrac {4!} {2! \times 2!}\)

Definition of Binomial Coefficient

\(\ds \)

\(=\)

\(\ds \dfrac 1 3 \times \dfrac {24} {2 \times 2}\)

Definition of Factorial

\(\ds \)

\(=\)

\(\ds \dfrac 1 3 \times 6\)

\(\ds \)

\(=\)

\(\ds 2\)

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $42$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $42$