2.5. Question 9: Stirling Numbers

Try these first

1. 1.

   What does S(3, 2) represent?

   Hint

   Think about how many ways you can create groups.

2. 2.

   Explain what a surjective function is in your own words.

   Hint

   Consider how every target element must be hit.

3. 3.

   What do Stirling numbers count?

   • Number of subsets
   • Number of ways to partition a set
   • Unique elements in a set

   Hint

   They're not just counting subsets; think broader!

4. 4.

   True or False: S(m, 1) is always 1.

   • True
   • False

   Hint

   Consider the meaning of one subset.

5. 5.

   Prove the recurrence relation by using specific examples of small sets and their partitions.

   Hint

   Break down each set and categorize!

6. 6.

   Calculate S(7, 5) by applying the recurrence relation stepwise until you reach known values.

   Hint

   List your calculations clearly!