$n$.

\[lemma.tower\] Let $p$ be a prime. Then, $$v_p({{{H}}_{}\!\left(\lambda\right)}) = \sum_{d=1} d \sum_{(q_i) \in \{0,\cdots,p-1\}^d }  \left|\lambda^{[q_1,\cdots,d_d]}\right|.
\label{eqn.valuation}$$

The proof of this lemma is an easy consequence of the stronger statement that for $d$ a positive integer $$\left|\left\{\square \in \lambda: v_p(h(\square))=d\right\}\right| =  \sum_{(q_i) \in \{0,\cdots,p-1\}^d }  \left|\lambda^{[q_1,\cdots,d_d]}\right|,
\label{eqn.stronger}$$ which itself results from the recursive definition of the $p$-core tower and Equation  for the base case.

**Remark.** If $p$ is not prime, remains true, but fails to imply . Consider the example of $p=6$ and $\lambda = (6)$. In that case, the $6$-core tower of $(6)$ consists of a single nontrivial label, by the partition of size 1, at a a node adjacent to the root (so the RHS evaluates to 1). Meanwhile, we have ${{{H}}_{}\!\left(\lambda\right)}=720$, so the LHS of evaluates to 2. This extra factor of 6 actually comes from the presence of hooks of length 2 and 3, with no direct consequence on the $6$-core tower.

Proofs of Theorem \[thm.landaugeneral\] and \[thm.multinomial\]
===============================================================

We immediately prove Theorem \[thm.landaugeneral\], the generalization of Landau’s theorem.

**$\Rightarrow$ :** We prove the statement by contraposition. Assume that for some $\mu$ the sum in is negative, *i.e. *$\sum_i {{h}_{r_i}\!\left(\mu\right)} - \sum_j {{h}_{s_j}\!\left(\mu\right)} < 0.$ 