Moving a bead one step upward on an $e$-runner <abacus of a partition> removes an $e$-hook and decreases the size by $e$. Sliding all beads upward reaches the <core of a partition> after exactly $w_e(\lambda)$ moves. Therefore
$$
|\lambda|=|C_e(\lambda)|+e\,w_e(\lambda).
$$

Solution

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

 Solution

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