and $t=t'g$, and let ${\mathcal Quo}_g({\lambda})=({\lambda}_1, \, \ldots , \, {\lambda}_g)$ be the $g$-quotient of ${\lambda}$. Then ${\lambda}$ is an $(s,t)$-core if and only if ${\lambda}_1, \, \ldots , \, {\lambda}_g$ are $(s',t')$-cores.

They obtain, as a corollary, a generating function $\psi_{s,t}$ for the number of $(s,t)$-core partitions of $n$.

[@AKS Corollary 4.3] \[genfunstcore\] With the above notation, we have $$\psi_{s,t}(x)= \psi_{s',t'}(x^g)^g  F_g(x) =\psi_{s',t'}(x^g)^g  \prod_{n=1}^{\infty}\frac{(1-x^{gn})^g}{1-x^n},$$ where $F_g$ is the generating function for the number of $g$-core partitions.

They also find the generating function for the number of self-conjugate $(s,t)$-cores when $g>1$.

[@AKS Theorem 7.1] \[genfunselfstcore\] The generating function $\psi^*_{s,t}$ for the number of self-conjugate $(s,t)$-cores with $g>1$ is given by $$\psi^*_{s,t} (x)= \left\{ \begin{array}{ll}F^*_{g}(x)(\psi_{s',t'}(x^{2g}))^{\frac{g}{2}} &  \mbox{if g is even,}\\
F^*_{g}(x)(\psi_{s',t'}(x^{2g}))^{\frac{g-1}{2}}\psi^*_{s',t'}(x^g) &  \mbox{if g is odd.}
\end{array}
\right.$$

Aukerman, Kane, and Sze also obtain a result on the number of $(s,t)$-core partitions which are not $g$-cores, and the existence of a $t$-core partition of $n$ that is not a $g$-core.

[@AKS Corollary 4.4] \[infinitelymanystcores\] If $g>1$, $s'>1$ and $t'>1$, then there are infinitely many $(s'g,t'g)$-cores which are not $g$-cores.

\[yeahtnog\] [@AKS Corollary 1.2] Let $g \geq 4$ be an integer. Then, for any integers $n \geq g$ and $t'>1$, there exists 