\[hookbij\] For any integer $k \geq 1$, there is a canonical bijection between the set of hooks of length $kt$ in ${\lambda}$ and the set of hooks of length $k$ in ${\mathcal Quo}_t({\lambda})$, where a hook in the $t$-quotient is simply a hook in any of ${\lambda}_1, \, \ldots , \, {\lambda}_t$.

Let $s=s'g$ and $t=t'g$, with $g>1$. We are now provide new, succinct proofs of Theorem \[stcore\] and Corollary \[genfunstcore\]. We note that the approach here provides the blueprint for many of the results that follow.

Recall that ${\lambda}$ is an $s$-core if and only if ${\lambda}$ has no hook of length divisible by $s=s'g$. Now, by Proposition \[hookbij\], there is a bijection between the set of hooks of length $ks=ks'g$ in ${\lambda}$ and the set of hooks of length $ks'$ in ${\mathcal Quo}_g({\lambda})$ for any $k \geq 1$. Hence ${\lambda}$ is an $s$-core if and only if ${\lambda}_1, \, \ldots , \, {\lambda}_g$ are $s'$-cores. Similarly, ${\lambda}$ is a $t$-core if and only if ${\lambda}_1, \, \ldots , \, {\lambda}_g$ are $t'$-cores. The result follows.

We write $Q_{s',t'}(g,w)$ for the number of $g$-tuples of $(s',t')$-cores of weight $w$, and $\hat{\psi}_{s,t}(n)$ for the number of $(s,t)$-cores of an integer 