Number Partitions

In mathematics, a partition of a number $ N $ is a set of numbers (less than or equal to $ N $) whose sum is $ N $.

The partition counting function $ p(n) $ counts the number of partitions of an integer $ 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 for a list of given coins/notes.

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

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

— Partitions into even parts: All the terms of the partition are even.

— Partitions into odd parts: All the terms of the partition are odd.

This is a partition where all terms are identical.

These partitions are predictable by knowing the list of divisors of the number.