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Number Partitions

Tool to generate and explore integer partitions. Discover in detail the decomposition of any number N into a set of smaller numbers, whose sum is equal to N.

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Number Partitions -

Tag(s) : Arithmetics

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# Number Partitions

## Partitions Count/Enumeration

### What is a partition of an integer number? (Definition)

Definition: in mathematics, a partition $p(N)$ of a number $N$ is a set of numbers (less than or equal to $N$) whose sum is $N$.

Example: The number $5$ can be decomposed into $7$ distinct partitions, the additions are: $5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1$

Permutations are ignored: $4+1$ and $1+4$ are considered identical

Example: The number $10$ has $42$ partitions/decompositions, and the number $100$ has $190569292$.

### What is the Hardy-Ramanujan formula?

In 1918, Hardy and Ramanujan have found an approximation of $p(n)$ for big numbers $n$ :

$$p(n) \sim \frac{1}{4n \sqrt{3}} ~ e^{\pi \sqrt{\frac{2n}{3}}}$$

### How to list Coin Change-making problem solutions?

Partitions of a number are used to solve the change-making problem and to list the ways of give back money.

### What are Ramanujan congruences?

The Ramanujan congruences, discovered by the mathematician Srinivasa Ramanujan, are particularly remarkable congruences that concern the partition function p(n).

\begin{align} p(5k+4) & \equiv 0 \pmod{5} \\ p(7k+5) & \equiv 0 \pmod{7} \\ p(11k+6) & \equiv 0 \pmod{11} \end{align}

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Number Partitions on dCode.fr [online website], retrieved on 2024-09-13, https://www.dcode.fr/partitions-generator

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