Character degree coprime to p from the core tower ($p\nmid {\chi }^{\lambda }(1)$)

Let $p$ be a prime number, let $n={\sum }_r{\alpha }_r{p}^r$ with $0\le {\alpha }_r<p$, and let $c_r$ be the sum of partition sizes at level $r$ of the core tower of a partition $\lambda ⊢n$. ThenIndeed $n={\sum }_r{c}_r{p}^r$. Carrying $p$ units from position $r$ to position $r+1$ decreases ${\sum }_r{c}_r$ by $p-1$. Normalization to the base-$p$ digits therefore gives ${\sum }_r{c}_r\ge {\sum }_r{\alpha }_r$, 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 $p-1$.