gen-briefly-explain-partitions-of-an-integer-and-one-common-misc-20c1

Briefly explain partitions of an integer and one common misconception about it.

# Partitions of an Integer **Definition:** A partition of a positive integer n is a way of writing n as a sum of positive integers, where the order doesn't matter. For example, the partitions of 4 are: - 4 - 3 + 1 - 2 + 2 - 2 + 1 + 1 - 1 + 1 + 1 + 1 So there are 5 partitions of 4. ## Common Misconception **Misconception:** Many people initially think partitions and compositions are the same thing. **Reality:** In partitions, **order doesn't matter** (3+1 and 1+3 are the same partition), while in compositions, **order does matter** (3+1 and 1+3 are different compositions). This means the number of compositions is always greater than or equal to the number of partitions. For n=4, there are 5 partitions but 8 compositions.