What is an integer partition?

An integer partition of n is a way of writing n as a sum of positive integers where order does not matter. For example, 4 has 5 partitions: 4, 3+1, 2+2, 2+1+1, 1+1+1+1.

p(n) is the partition function — the total count of distinct partitions of n. It grows rapidly: p(10) = 42, p(50) = 204,226, p(100) = 190,569,292.

How is p(n) computed?

This calculator uses Euler's pentagonal number recurrence: p(n) = Σ (-1)^(k+1) p(n - k(3k-1)/2) summed over all pentagonal numbers k = ±1, ±2, …, with p(0)=1. It runs in O(n√n) time and handles large n precisely using BigInt arithmetic.

Why are partitions not listed for large n?

The number of partitions grows very fast (p(20) = 627, p(30) = 5604). Listing them all for large n would flood the page; the count is shown for any n up to 500.

Does order matter in partitions?

No. 3+1 and 1+3 are the same partition. If order mattered they would be called compositions, and the count would be 2^(n-1) instead.

Partition Number Calculator

Name: Partition Number Calculator
Availability: InStock
Author: Nham Vu

Enter a non-negative integer to compute p(n) — the count of ways to write it as an ordered sum — and browse all partitions for n ≤ 20.

Enter an integer n

n (0 – 500)

Partitions listed for n ≤ 20

Quick examples

Result

Enter a value and click Compute.

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Summary

Enter a non-negative integer to compute p(n) — the count of ways to write it as an ordered sum — and browse all partitions for n ≤ 20.

How it works

Enter a non-negative integer n in the input field.
The calculator computes p(n) using Euler's recurrence relation in O(n√n) time.
For n ≤ 20, every distinct partition is listed (e.g. 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1).
Above n = 20, only the count p(n) is shown — the number of partitions grows too fast to list.
Use the examples panel to jump to common values and explore the sequence.

Use cases

Check your combinatorics homework answer for partition numbers.
Explore the partition function sequence p(0)–p(n) in number theory.
Count the number of ways to break a number into additive parts for puzzles.
Understand partitions in the context of Young tableaux and representation theory.
Verify a specific partition listing for a small integer by hand.
Teach integer partitions interactively in a discrete math course.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu