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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Number Partitions
Partitions Calculator
Partitions Count/Enumeration p(N)
Answers to Questions (FAQ)
What is a partition of an integer number? (Definition)
In mathematics, a partition 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 $
By default, partitions are composed only of non-zero natural integers.
What is a the function p(N)?
The partition counting function $ p(n) $ counts the number of partitions of an integer $ n $.
Permutations of partitions are not counted: $ 4+1 $ and $ 1+4 $ are considered identical
Example: The number $ 10 $ has $ 42 $ partition decompositions, $ p(10) = 42 $, and $ p(100) = 190569292 $
What is the Hardy-Ramanujan formula?
In 1918, Hardy and Ramanujan have found an approximation of $ p(n) $ for big numbers $ n $ :
$$ p(n) \sim \frac{1}{4n \sqrt{3}} ~ e^{\pi \sqrt{\frac{2n}{3}}} $$
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 for a list of given coins/notes.
Example: There are 49 ways to make $100 with $5, $10, $20 or $50 notes
What is the difference between distinct and non-distinct partitions of a number?
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
What is a Ferrers diagram?
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.
Example: The partition $ 5 = 3 + 2 $ can be represented
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What are Ramanujan congruences?
The Ramanujan congruences, discovered by the mathematician Srinivasa Ramanujan, are particularly remarkable congruences that concern the partition function p(n).
$$ \begin{align} p(5k+4) & \equiv 0 \pmod{5} \\ p(7k+5) & \equiv 0 \pmod{7} \\ p(11k+6) & \equiv 0 \pmod{11} \end{align} $$
What is an even (or odd) partition?
— Partitions into even parts: All the terms of the partition are even.
— Partitions into odd parts: All the terms of the partition are odd.
Example: For $ n = 4 $, the partitions into even parts are $ 4, ; 2+2 $, while there is no partition into odd parts.
What is an equal partition?
This is a partition where all terms are identical.
Example: For $ n = 6 $, the only partition in equal parts is $ 2+2+2 $
These partitions are predictable by knowing the list of divisors of the number.
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- Partitions Calculator
- Partitions Count/Enumeration p(N)
- What is a partition of an integer number? (Definition)
- What is a the function p(N)?
- 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 are Ramanujan congruences?
- What is an even (or odd) partition?
- What is an equal partition?
