gen-in-3-4-sentences-explain-partitions-of-an-integer-state-any-639a

In 3-4 sentences, explain partitions of an integer. State any key formula.

A partition of a positive integer n is a way of writing n as a sum of positive integers, where the order of the summands doesn't matter. For example, the partitions of 4 are: 4, 3+1, 2+2, 2+1+1, and 1+1+1+1, giving 5 total partitions. The number of partitions of n is denoted p(n), and while there's no simple closed formula, the **generating function** is given by: $$\sum_{n=0}^{\infty} p(n)x^n = \prod_{k=1}^{\infty} \frac{1}{1-x^k}$$ This elegant formula encodes all partition numbers in a single infinite product.