Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 160 / 3 / b / Solution

Using the abacus of a partition in James's convention, read runner $r$ as a one-runner abacus: replace each occupied position $r+eq$ by $q$. After the usual harmless shift of its beta set, the resulting partition is ${\lambda }^{(r)}$. Doing this for $r=0,\dots ,e-1$ produces

Conjugating a partition complements beads and gaps and reflects the abacus. Reflection sends runner $r$ to runner $e-1-r$, reverses bead-gap order, and conjugates the runner partition. Hence