\emptyset)$. Then ${\lambda}_0$ is an $(\bar{s'},\bar{t'})$-core (since $s',t'\neq1$), and ${\lambda}_1, \, \ldots , \, {\lambda}_{\frac{g-1}{2}}$ are certainly $(s',t')$-cores. Thus, by Theorem \[stbarcore\], ${\lambda}$ is a $(\overline{s'g},\overline{t'g})$-core (note here that Theorem \[stbarcore\] does not require $g=\text{gcd}(s'g,t'g)$). Also, since ${\mathcal Quo}_{\bar{g}}({\lambda}) \neq (  \emptyset, \, \ldots , \, \emptyset)$, the bar-partition ${\lambda}$ is certainly not a ${\bar{g}}$-core. Allowing ${\gamma}$ to vary produces an infinite number of (distinct) $(\overline{s'g},\overline{t'g})$-cores which are not $\bar{g}$-cores.

For odd integers $g$ and $t'$, let $Q_{\bar{t}'}(g,w)$ be the number of ${\bar{g}}$-quotients $({\lambda}_0, \ldots , {\lambda}_{\frac{g-1}{2}})$ of weight $w$ such that ${\lambda}_0$ is a $\bar{t}'$-core and ${\lambda}_1, \, \ldots , \, {\lambda}_{\frac{g-1}{2}}$ are $t'$-cores.

\[tbarnotg\] Let $g \geq 7$ and $t'>1$ be odd integers, and let $t=t'g$. Then, for any $n \geq g$, the number $\hat{\xi}_{\bar{t}\backslash {\bar{g}}}(n)$ of $\bar{t}$-core partitions of $n$ which are not $\bar{g}$-cores satisfies $$\hat{\xi}_{\bar{t}\backslash {\bar{g}}}(n) \geq {\displaystyle}\sum_{w= 1}^{\left\lfloor \frac{n}{g} \right\rfloor} Q_{\bar{t}'}(g,w).$$ In particular, $\hat{\xi}_{\bar{t}\backslash {\bar{g}}}(n) \geq \frac{g+1}{2}$ and there exists a $\bar{t}$-core partition of $n$ which is not a ${\bar{g}}$-core.

Note that a bar-partition ${\lambda}$ is a ${\bar{g}}$-core if and only if it has ${\bar{g}}$-weight 0. We thus have $$\hat{\xi}_{\bar{t}\backslash {\bar{g}}}(n)  =  {\displaystyle}\sum_{w\geq 1} Q_{\bar{t}'}(g,w) f_{{\bar{g}}}(n-gw) =  {\displaystyle}\sum_{w= 1}^{\left\lfloor \frac{n}{g} \right\rfloor} Q_{\bar{t}'}(g,w) f_{{\bar{g}}}(n-gw)  \geq  {\displaystyle}\sum_{w= 