proof about one of the claims in Partition

Question

Can anyone explain or prove to me why the claim of partition that the "number of partitions of $n$ is equal to the number of partitions of $2n$ with $n$ parts" is true, thanks.

Answer 1

If we are given a partition of $2n$ into $n$ parts, each part is positive by definition. If we subtract $1$ from each part, we get a list of numbers that add up to $2n-n=n$. Some of these numbers will be $0$, when the corresponding part in the original partition was $1$, but if we discard the $0$'s we get a partition of $n$.

Conversely, start with a partition of $n$ into $k$ parts. Add $1$ to each part and adjoin $n-k$ $1$'s to get a partition of $2n$. It is clear that the two maps are inverses of one another, so this is a bijection.

Example: $n=6$

$$6=2+4$$ is a partition of $6$ into $k=2$ parts. To get a partition of $12$ into $4$ parts we add $1$ to each part and adjoin $4\space1$'s:$$ 12=1+1+1+1+3+5$$

Conversely, if we started with the partition of $12,$ subtracted $1$ from each part, and ignored the resulting $0$'s, we would get the original partition of $6$ back.