a $p$-tuple of $p$-quotients for $\lambda$. Figure \[fig.01\] presents a computation of quotients.

These quotient partitions have the property, essential for us, that $$\label{eqn.essential}
{{\mathcal{H}}_{p}\!\left(\lambda\right)} = \underset{0 \le j \le p-1}{\cup}  {{\mathcal{H}}_{}\!\left(\lambda^{(j)}\right)},$$ with the union taken with multiplicity (Figure \[fig.hooks\] has the hook lengths for the partition that appears in Figure \[fig.01\] and its quotients). In other words, the union of the hook lengths of the $p$-quotients of $\lambda$ give the hook lengths of $\lambda$ that are divisible by $p$, divided by $p$. In particular, this implies ${{h}_{p}\!\left(\lambda\right)} = \sum_{0 \le j \le p-1} {{h}_{}\!\left(\lambda^{(j)}\right)}$ or even ${{h}_{pk}\!\left(\lambda\right)} = \sum_{0 \le j \le p-1} {{h}_{k}\!\left(\lambda^{(j)}\right)}$.

For each positive integer $p$, we have so far defined a map, $
\{\text{partitions} \} \rightarrow \{ p-\text{cores} \} \times \{ \text{partitions} \}^p,
$ called the *Littlewood decomposition at $p$*, and the construction actually shows that this map is a bijection. This can be thought of as a generalization of Euclidian division for integers. We refer the reader to [@Macdonald] for more information. The Littlewood decomposition also satisfies $$\label{eqn.Littlewood_counts}
|\lambda| = |p-\text{core}(\lambda)| + p \sum_{j=0}^{p-1} |\lambda^{(j)}|,$$ which will be useful for us (we repeat that there is no requirement for $p$ to be prime).

Integers 