Parenthesization/Examples/2

Example of Parenthesization

A word of $2$ elements can be parenthesized in only $1$ distinct way:

$\quad \paren {a_1 a_2}$

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 $n = 2$ we have:

\(\ds C_1\)

\(=\)

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

\(\ds \)

\(=\)

\(\ds \dfrac 1 2 \times 2\)

Binomial Coefficient with One

\(\ds \)

\(=\)

\(\ds 1\)

$\blacksquare$

Sources

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