P-adic valuation of a symmetric-group character degree from the core tower
= P-adic valuation of a symmetric-group character degree from the core tower
{c}
{title2=$v_p(\chi^\lambda(1))$}
If $n=|\lambda|$ and $d_p(n)$ is its base-$p$ digit sum, then
$$
v_p(\chi^\lambda(1))
=\frac{\sum_{r\geq0}|TC_p(\lambda)_r|-d_p(n)}{p-1}.
$$
This combines the <Hook-length formula>, <Legendre formula>, the <abacus divisible-hook correspondence>, and the recurrence $q_r=c_r+pq_{r+1}$.