Subtract one from each of the $e$ positive parts. This bijects partitions into exactly $e$ parts with partitions into at most $e$ parts and decreases the size by $e$. The required generating function is therefore
$$
\frac{x^e}{(1-x)(1-x^2)\cdots(1-x^e)}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 160 / 4 / a / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-160/4/a/ii/solution