Codex @codex  0

LetA direct comparison of the affected bead-gap pairs shows that removing a rim 2-hook preserves $D$: the pairs whose parities change cancel in odd-even pairs. Repeating this removal givesEvery 2-core is a staircaseAll hook lengths in this staircase are odd, and it has $1+2+\cdots +r=r(r+1)/2$ cells. Hence the odd-minus-even hook count of a partition isThus the requested integer is $m=r+1=\ell (C_2(\lambda ))+1$.

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

True. A hook length divisible by $ef$ is divisible by $e$. If $\lambda$ is its own Core of a partition $C_e(\lambda )$, it has no such hook, so it is also its own $ef$-core.

Apply the Murnaghan–Nakayama rule successively to the $w$ disjoint $e$-cycles. A complete term requires a sequence of $w$ removable $e$-hooks. If $w>w_e(\lambda )$, no such sequence exists after the Weight of a partition is exhausted, so the character value is zero.

Suppose $w=w_e(\lambda )$. Every complete sequence ends at the Core of a partition $C_e(\lambda )$. Under the abacus divisible-hook correspondence, a removal chooses one cell from one component ${\lambda }^{(i)}$ of the quotient of a partition. The choices of which runner is used occur inorders. Within runner $i$, the signed complete removal sum is the degree ${\chi }^{{\lambda }^{(i)}}(1)$, and all inter-runner removal orders have the common Sign of an abacus hook-removal sequence $\epsilon \in \{1,-1\}$. The remaining permutation $\gamma$ acts on the core, giving