Overview
This lecture explains the problem of counting integer partitions using recursion, focusing on the recursive strategy and implementation for the count_partitions function.
Counting Integer Partitions
- A partition of n using parts up to size m is a way to express n as a sum of positive integers ≤ m in increasing order.
- For example, count_partitions(6, 4) counts all ways to write 6 using parts of size 1, 2, 3, or 4, in increasing order.
- The function does not return the partitions themselves, but the number of possible partitions.
Recursive Decomposition Strategy
- The problem can be divided into two cases: partitions including at least one m, and partitions with only smaller parts (< m).
- If at least one m is used, partition n - m using parts up to size m.
- If no m is used, partition n using parts up to size m - 1.
- The total number of partitions is the sum of these two cases.
- This recursive approach creates a tree recursion structure, exploring all possibilities.
Base Cases and Implementation
- Base case 1: If n == 0, there is one way to partition (use no parts).
- Base case 2: If n < 0, there are zero ways to partition (invalid partition).
- Base case 3: If m <= 0, there are zero ways to partition (no valid parts left).
- Implementation uses the recursive approach by summing the two cases and handling the base cases.
Example: count_partitions(5, 3)
- The function explores all ways to partition 5 using 1, 2, or 3 as part sizes.
- Recursive calls break the problem into simpler subproblems until reaching base cases.
- The sum of valid alternatives gives the total number of partitions.
Key Terms & Definitions
- Partition — An expression of a number as a sum of positive integers in increasing order.
- Tree Recursion — A recursive structure where each call can branch into multiple subcalls, used here to handle alternative partition choices.
Action Items / Next Steps
- Review the recursive decomposition for count_partitions.
- Practice implementing count_partitions in code.
- Try examples such as count_partitions(6, 4) and count_partitions(5, 3) to solidify understanding.