20.1. Catalan Numbers

Try these first

1. 1.

   What is the value of C(0)?

   Hint

   Think of an empty sequence.

2. 2.

   How many ways can you parenthesize 3 numbers?

   Hint

   Use C(2) as your target.

3. 3.

   What is the nth Catalan number C(n) primarily used for?

   • Counting binary trees
   • Finding prime numbers
   • Calculating Fibonacci numbers

   Hint

   Relate it to tree structures.

4. 4.

   The formula C(n) = Σ C(k) * C(n-k-1) helps in which context?

   • True
   • False

   Hint

   Think of it as breaking down arrangements.

5. 5.

   Consider a sequence of expressions involving n pairs of parentheses. Determine how many unique arrangements can be made without altering the original order of pairs based on the properties of Catalan numbers.

   Hint

   Break arrangements down recursively as structured pairs.

6. 6.

   Explore another combinatorial problem involving non-crossing handshakes. Calculate how many valid configurations there are by relating them to Catalan numbers.

   Hint

   How many handshakes yield a non-crossing configuration?