Tool to generate and explore integer partitions. Discover in detail the decomposition of any number N into a set of smaller numbers, whose sum is equal to N.

Number Partitions - dCode

Tag(s) : Arithmetics

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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 $.

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

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

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

Distinct partitions of an integer are partitions where the integers in the sum are all distinct from each other.

__Example:__ 5 = 1+4 = 2+3

Non-distinct partitions include repeated numbers.

__Example:__ 5 = 1+1+1+2 = 1+2+2

Ferrers diagrams are graphical representations of the partitions of a number using dots or boxes in rows.

Each row represents a number in the partition sum. Ferrers diagrams are a visual way to study the partitions of a number and understand their structure.

The partitions function, often denoted $ p(n) $, is a mathematical function that counts the number of distinct ways of partitioning a positive integer $ n $ into a sum of positive integers, regardless of the order of the terms. In other words, $ p(n) $ gives the number of different partitions of a given number $ n $.

The Ramanujan congruences, discovered by the mathematician Srinivasa Ramanujan, are particularly remarkable congruences that concern the partition function p(n).

$$ \begin{align} p(5k+4) & \equiv 0 \pmod{5} \\ p(7k+5) & \equiv 0 \pmod{7} \\ p(11k+6) & \equiv 0 \pmod{11} \end{align} $$

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Cite as source (bibliography):

*Number Partitions* on dCode.fr [online website], retrieved on 2024-09-13,

- Partitions Calculator
- Partitions Count/Enumeration
- What is a partition of an integer number? (Definition)
- What is the Hardy-Ramanujan formula?
- How to list Coin Change-making problem solutions?
- What is the difference between distinct and non-distinct partitions of a number?
- What is a Ferrers diagram?
- What is the partition function p(n)?
- What are Ramanujan congruences?

partition,decomposition,sum,set,number,integer,ferrers

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