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.

Number Partitions - dCode

Tag(s) : Mathematics, Arithmetics

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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.

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

Example: The number \( 5 \) can be decomposed into \( 7 \) distinct partitions : \( 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1 \)

Permutations are ignored: \( 4+1 \) and \( 1+4 \) are considered identical

Example: The number \( 10 \) has \( 42 \) partitions, and the number \( 100 \) has \( 190569292 \).

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

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

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

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

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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Source : https://www.dcode.fr/partitions-generator

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