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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) : Mathematics, Arithmetics

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Number Partitions

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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.

Answers to Questions

What is a partition of an integer number?

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 : \( 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1 \)

Permutationshref 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.

What is the Hardy-Ramanujan formula?

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

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

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

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