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Number Partitions

Tool to generate partitions of a number (integer). A partition of an integer N is a decomposition of N into a set of numbers (inferior to N) which sum is 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 $190569292$.

Due to servers' computation's cost with large lists, free generations are limited.

### 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.

Example: There are 49 ways to make $100 with$5, $10,$20 or \$50 notes

### How to remove the limits of the generator?

The generation is very costful in resources (which are expensive) as soon as the quantity of solution becomes large. dCode offers exhaustive lists, ask for prices!

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Cite as source (bibliography):
Number Partitions on dCode.fr [online website], retrieved on 2022-12-09, https://www.dcode.fr/partitions-generator

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