P-adic valuation of a symmetric-group character degree from the core tower ($v_p({\chi }^{\lambda }(1))$)

If $n=|\lambda |$ and $d_p(n)$ is its base-$p$ digit sum, thenThis combines the Hook-length formula, Legendre formula, the abacus divisible-hook correspondence, and the recurrence $q_r=c_r+p{q}_{r+1}$.

For every partition $\lambda$,The core-tower formula reduces this to subadditivity of the base-$p$ digit sum across the sizes of the quotient partitions.