Calculator Form
Example Data Table
| Target Integer | Example Four-Part Partition | Multiplicity | Product | Range |
|---|---|---|---|---|
| 12 | 1 + 1 + 1 + 9 | 4 | 9 | 8 |
| 12 | 1 + 1 + 5 + 5 | 6 | 25 | 4 |
| 12 | 1 + 2 + 4 + 5 | 24 | 40 | 4 |
| 12 | 2 + 2 + 4 + 4 | 6 | 64 | 2 |
Formula Used
The calculator searches all nondecreasing four-part tuples (a, b, c, d) that satisfy:
- a + b + c + d = n
- a ≤ b ≤ c ≤ d for unique partitions
- a < b < c < d when distinct parts are required
- Optional minimum and maximum bounds are applied to every part
Ordered arrangements are counted from each unique row with: 4! ÷ (m1! × m2! × ...), where each m is the repeat count of equal values.
Product is found with a × b × c × d. Range is found with d − a. These extra metrics help compare balanced and unbalanced partitions.
How to Use This Calculator
- Enter the target integer you want to split.
- Set the minimum part. Use zero only when needed.
- Add an optional maximum part to limit the search.
- Choose whether all four parts must be distinct.
- Select how many rows you want displayed.
- Choose a sorting rule for the result table.
- Press the calculate button to view counts and rows.
- Use the export buttons to save CSV or PDF copies.
Frequently Asked Questions
1. What does four partitions mean here?
It means the target integer is split into exactly four parts. The calculator lists valid four-value combinations that match your limits.
2. Why are rows written in nondecreasing order?
Nondecreasing order prevents duplicate rows. For example, 1 + 2 + 4 + 5 and 5 + 4 + 2 + 1 represent the same partition.
3. What is multiplicity?
Multiplicity counts how many distinct orderings belong to one partition row. A row with repeated values has fewer arrangements than a row with four different values.
4. Can I allow zero in a partition?
Yes. Tick the zero option and use a minimum part of zero. Then rows like 0 + 2 + 3 + 7 can be included.
5. What happens when distinct parts are required?
The calculator applies strict inequality. Every valid row must have four different values, so repeated parts are excluded automatically.
6. Why is there a maximum target value?
The cap keeps the page responsive. Exact partition searches can grow quickly, especially when many result rows are possible.
7. What does the graph show?
The graph shows how many valid four-part partitions begin with each smallest part. It is useful for seeing how solutions spread across the search space.
8. Can I save the results?
Yes. Use the CSV button for spreadsheet work. Use the PDF button when you need a printable report of the visible results.