21.1.11. General Process of S' Construction

Try these first

1. 1.

   Define a valid sequence.

   Hint

   Think about sequences of +1 and -1.

2. 2.

   What is the cardinality of all sequences of n +1s and –1s?

   Hint

   Consider how many ways to arrange +1s and -1s.

3. 3.

   What defines a valid sequence?

   • A sequence containing more -1s
   • A sequence where the sum is always non-negative
   • Any sequence of +1s and -1s

   Hint

   Focus on the non-negativity rule.

4. 4.

   True or False: The reflection method can convert any sequence into a valid one.

   • True
   • False

   Hint

   What happens during the reflection process?

5. 5.

   Explain how to derive the closed form of Catalan numbers from first principles. Use examples with n=2.

   Hint

   Consider the sequences and apply the reflection method to count.

6. 6.

   Prove the bijective nature of the reflection method.

   Hint

   Use sequence properties to highlight uniqueness.