A partition of a positive integer $n$ is to write $n$ as a sum of some positive integers. Let $k$ be a positive integer. Show that the number of partitions of $n$ with exactly $k$ parts equals the number of partitions of $n$ whose largest part is exactly $k$.
This claim can be explained by visualization. Consider a partition of $n$ as $$n=a_1 + a_2 + \cdots + a_k$$
where $a_1 \ge a_2 \ge \cdots \ge a_k$. This partition can be represented by an array of dots. The left part of the following diagram illustrates an example of $10=5+3+1+1$. Then, this diagram can flipped along its main diagonal to create a paired partition whose largest part equals the number of partitions in the original partition.
Meanwhile, the right side of the diagram can also be mapped back to the left side. Hence, they have a one-to-one mapping relationship. This means the claim holds.