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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 Calculator


Partitions' Size






Decomposition Content





Partitions Count/Enumeration


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 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 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

How 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 !

Source code

dCode retains ownership of the online 'Number Partitions' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Number Partitions download for offline use on PC, tablet, iPhone or Android !

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Questions / Comments

Thanks to your feedback and relevant comments, dCode has developped the best Number Partitions tool, so feel free to write! Thank you !


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