= Maximal p-hook character formula
{title2=$\chi^\lambda(\pi p^{w_p(\lambda)})=\varepsilon_p(\lambda)N_p(\lambda)\chi^{C_p(\lambda)}(\pi)$}
Let $\kappa$ be the <core of a partition> $\lambda$ at prime modulus $p$, with quotient component sizes $w_i$ and total weight $w$. Complete $p$-hook removal sequences number $N_p(\lambda)=\binom{w}{w_0,\ldots,w_{p-1}}\prod_if^{\lambda^{(i)}}$. Their signs agree because abacus bead-crossing parities telescope to the <permutation> taking the initial labeled beads to the packed core order. Repeated <Murnaghan–Nakayama rule> gives the displayed formula for any <permutation> $\pi$ on the remaining letters.
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