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
$$
\prod_{j=1}^\infty(1+x^{2j-1}).
$$

Solution

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

A self-conjugate partition is determined by the hook lengths of its diagonal cells. These are distinct odd positive integers, and every partition into distinct odd parts arises uniquely this way. Hence their generating function is
$$
\prod_{i=1}^\infty(1+x^{2i-1}).
$$


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