Number Partitions

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 $.

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.

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

Non-distinct partitions include repeated numbers.

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).

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