the generating function for the number of ${\bar{g}}$-core partitions.

Each $(\bar{s},\bar{t})$-core partition ${\lambda}$ of $n$ is completely and uniquely determined by its ${\bar{g}}$-core ${\mathcal Cor}_{{\bar{g}}}({\lambda})$ and ${\bar{g}}$-quotient ${\mathcal Quo}_{\bar{g}}({\lambda})$. And, by Theorem \[stbarcore\], if ${\lambda}$ has ${\bar{g}}$-weight $w$, then ${\mathcal Quo}_{{\bar{g}}}({\lambda})$ can be any ${\bar{g}}$-quotient $({\lambda}_0, \, {\lambda}_1, \, \ldots , \, {\lambda}_{\frac{g-1}{2}})$ of weight $w$ such that ${\lambda}_0$ is an $(\bar{s}', \bar{t}')$-core and ${\lambda}_1, \, \ldots , \, {\lambda}_{\frac{g-1}{2}}$ are $(s',t')$-cores. Writing $Q_{\bar{s}',\bar{t}'}(g,w)$ for the number of such ${\bar{g}}$-quotients, this shows that the enumerating function $\hat{\xi}_{{\bar{s}},\bar{t}}(n)$ of $(\bar{s},\bar{t})$-cores of $n$ is $$\label{ok2}
\hat{\xi}_{{\bar{s}},\bar{t}}(n)={\displaystyle}\sum_{w\geq 0} Q_{\bar{s}',\bar{t}'}(g,w)f_{{\bar{g}}}(n-gw).$$ Viewed as a generating function,this transforms into $$\xi_{{\bar{s}},\bar{t}}(x)= \xi_{\bar{s'},\bar{t}'}(x^g)  \psi_{s',t'}(x^g)^{\frac{g-1}{2}}  F_{\bar{g}}(x),$$ as claimed.

We will return to these functions in Section 3 and again in Section 5.

Strengthening positivity for core partitions.
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In this section we consider $t$-cores and self-conjugate $t$-cores that are not $g$-cores, and $\bar{t}$-cores that are [*not*]{} $\bar{g}$-cores. We call these $t \backslash g$[*-cores*]{}, $t^* \backslash g$[*-cores*]{} and $\bar{t} \backslash \bar{g}$[*-cores*]{} respectively. We first study $(s,t)$-core partitions that are [*not*]{} $g$-cores and $(\bar{s}, \bar{t})$-cores that are [*not*]{} $\bar{g}$-cores. Again our convention is to fix positive integers $s$ and $t$ where $s=s'g$ and $t=t'g$, and $g=$gcd$(s,t)$.

Positivity for core partitions.
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