Working out partitions involves finding all the unique ways a positive integer can be expressed as a sum of other positive integers, where the order of the summands does not matter.
Understanding Partitions of an Integer
A partition of n is formally defined as a multiset of positive integers that add to n. This means you are looking for collections of positive whole numbers that, when summed together, equal n. A crucial point is that the order of the integers in the sum does not affect the partition; for instance, 1+2 is considered the same partition as 2+1.
The number of partitions for a given integer k is commonly denoted by p(k). For example, to "work out" the partitions of 3, we identify all unique ways to sum positive integers to 3.
Steps to Work Out Partitions (with Example)
To systematically work out partitions for a number n, you list all possible combinations of positive integers that sum up to n, ensuring no duplicates are included (since order doesn't matter).
Let's illustrate this process by working out the partitions of n = 3, as demonstrated in the reference:
- Start with the number itself: The simplest partition is the number itself.
3
- Consider sums involving the next largest possible integer: Try to include the integer
n-1, thenn-2, and so on, filling the remainder with smaller integers.- For
n=3, the next largest integer is2. 2 + 1(This is the same as 1+2)
- For
- Continue with sums involving smaller integers, ensuring parts are non-increasing: If you only use
1s, how many1s do you need?1 + 1 + 1
By following this systematic approach, we identify all unique partitions for 3.
Partitions of 3
| Partition | Sum |
|---|---|
| 3 | 3 |
| 2 + 1 | 3 |
| 1 + 1 + 1 | 3 |
As shown in the example, by listing these unique sums, we determine that the number of partitions of 3, denoted as p(3), is 3.
Key Considerations When Working Out Partitions
- Positive Integers Only: Partitions are formed using only positive whole numbers.
- Order Does Not Matter: Treat sums like
1+2and2+1as identical partitions. This is why they are often listed with parts in non-increasing order (e.g.,2+1rather than1+2) to avoid redundancy. - Systematic Listing: For smaller numbers, a methodical approach of starting with the largest possible parts and gradually decreasing them helps ensure all partitions are found without repetition.
While a manual listing is feasible for small numbers, working out partitions for larger integers becomes complex and typically involves more advanced combinatorial techniques or computer algorithms.
