Understand the subject
What is an integer partition?
An integer partition writes a number as positive parts while ignoring their order. The parts 3 + 1 and 1 + 3 are one partition.
Use it when only the multiset of positive parts matters.
See the structure
Order is deliberately ignored
43 + 12 + 22 + 1 + 11 + 1 + 1 + 1
Worked example
Check the rule with a small case
The five unordered positive-part sums of 4 mean p(4) = 5.
Interpret with care
Choose the right counting model
If 1 + 3 and 3 + 1 are distinct, use integer compositions instead.
Use the Permutation & Combination Generator when a small case needs to be inspected rather than only counted.
Quick guide
How to use this calculator
- Enter the integer n.
- Different orders of the same positive parts are treated as one partition.
- Read the partition number p(n).
Calculation method
Accumulate partitions by allowed part size
A dynamic program adds each possible positive part in increasing order, which counts every unordered partition once.
Worked example
Partition four
The partitions are 4, 3+1, 2+2, 2+1+1, and 1+1+1+1.
p(4) = 5
Supported inputs
Precision and limits
Discrete domain
Inputs are whole counts. Negative, fractional, grouped, and scientific-notation inputs are rejected.
Exact integers
Results use arbitrary-precision integers and are never rounded. Inputs and output size are bounded to keep the page responsive.
Maximum n
n is limited to 1,000 to keep the exact dynamic calculation responsive.
