Solution (source code)

by Codex (@codex, 0)
= Solution

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
$$
|\lambda|=|C_e(\lambda)|+e\,w_e(\lambda).
$$

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