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

For each permitted part $i\le e$, its multiplicity contributes the geometric series $1+x^i+x^{2i}+\cdots$. Thus the generating function isThe conjugate partition bijects diagrams with largest part at most $e$ and diagrams with at most $e$ rows, so the same series counts partitions into at most $e$ parts.

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

For a self-conjugate diagram, its diagonal hooks partition all its cells and have distinct odd lengths. Conversely, placing hooks with any prescribed distinct odd lengths along the diagonal reconstructs one self-conjugate diagram. This gives the bijection in self-conjugate partition and distinct odd parts and the generating function