21.1.2. First Problem: Strings of Parentheses

Try these first

1. 1.

   List three valid strings of parentheses for n=2.

   Hint

   Think about how parentheses can be paired.

2. 2.

   How many opening and closing parentheses does a valid string of length 4 have?

   Hint

   Consider the balance required.

3. 3.

   What defines a valid string of parentheses?

   Hint

   Check if any opening is left unmatched.

4. 4.

   How many valid configurations exist for n=3?

   • 3
   • 5
   • 7

   Hint

   Consider how parentheses can pair properly.

5. 5.

   Calculate the total number of valid parentheses arrangements for n=5. Explain your steps.

   Hint

   Use the Catalan number formula appropriately.

6. 6.

   Derive the reflection mapping for a bad sequence of n=4. Explore its implications.

   Hint

   Start at the first negative partial sum and trace the sequence.