= Character degree coprime to p from the core tower
{title2=$p\nmid\chi^\lambda(1)$}
Let $p$ be a <prime number>, let $n=\sum_r\alpha_rp^r$ with $0\leq\alpha_r<p$, and let $c_r$ be the sum of partition sizes at level $r$ of the <core tower of a partition> $\lambda\vdash n$. Then
$$
\boxed{p\nmid\chi^\lambda(1)\quad\Longleftrightarrow\quad c_r=\alpha_r\text{ for every }r.}
$$
Indeed $n=\sum_rc_rp^r$. Carrying $p$ units from position $r$ to position $r+1$ decreases $\sum_rc_r$ by $p-1$. Normalization to the base-$p$ digits therefore gives $\sum_rc_r\geq\sum_r\alpha_r$, with equality exactly when no carry is needed. The <P-adic valuation of a symmetric-group character degree from the core tower> is the difference divided by $p-1$.
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