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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

## 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.

### 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 : $$5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1$$

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.

### 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