# of partitions of $n$ into at most $r$ positive integers $=$ # of partitions of $n + r$ into exactly $r$ positive integers? How do I see that the number of partitions of the integer $n$ into at most $r$ positive integers is equal to the number of partitions of $n + r$ into exactly $r$ positive integers?

Consider a Ferrer's diagram of $n$ into at most $r$ parts.
$ooooooo\\
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o$
For example this is a partition of $17$ into $4$ parts - $17=7+6+3+1$.
So in this example we know $r\ge4$. This means we can insert a first column of size exactly $r$ to get a partition of $n+r$ with exactly $r$ parts.
We also need to establish a bijection, and we can note that, say $r=4$, then $17-4=13=(7-1)+(6-1)+(3-1)+(1-1)=6+5+2$, which can be said for all the partitions here.
Algebrically, the number of partitions of $n$ into at most $r$ part is given by the coefficient of $x^n$ in:
$$\prod_{k=1}^r\dfrac{1}{1-x^k}$$
Into exactly $r$ parts can be expressed as the coefficient of $x^{n+r}$ in:
$$x^r\prod_{k=1}^r\dfrac{1}{1-x^k}$$
