Prove that using generating function:For any $n ,k\in N$, the number of partitions of $n$ into parts

Question

For any $n,k\in N$, the number of partitions of $n$ into parts, each of which appears at most $k$ times, is equal to the number of partitions of $n$ into parts the sizes which are not divisible by $k+1$.

Answer 1

The generating function for partitions by parts each of which appears at most $k$ times is $$\prod_{q\ge 1} \sum_{m=0}^k x^{qm} = \prod_{q\ge 1} \frac{1-x^{q(k+1)}}{1-x^q}.$$ This is $$\prod_{q\ge 1} \frac{1}{1-x^q} \prod_{q\ge 1} \left(1-x^{q(k+1)}\right).$$ The first term here generates ordinary partitions and the second one cancels the parts that are a multiple of $k+1,$ QED.