which is not a $g$-core.

If $n <2g$, then this is all we can say, as the second sum is empty. If, on the other hand, $n \geq 2g$, then, for $w_1=1$ and $w_2=0$ (which does contribute to the second sum), we get $f_{t'}(w_1)=f_{t'}(1)=1$ and $f^*_{t'}(w_2)=f^*_{t'}(0)=1$ (corresponding to the empty partition). Hence, in this case, we have $\hat{\psi}_{t^*\backslash g}(n) \geq 1 + \frac{g-1}{2}= \frac{g+1}{2} \geq \frac{g}{2}$, as claimed.

If, furthermore, $t'=8$ or $t'\geq 10$, then, by Theorem \[Ono\] and Theorem \[fordsze\], $f_{t'}(w_1) \geq 1$ and $f^*_{t'}(w_2) \geq 1$ for all $w_1$ and $w_2$, and thus, in this case $$\hat{\psi}_{t^*\backslash g}(n) \geq 1 + {\displaystyle}\sum_{{\genfrac{}{}{0pt}{}{w_1 \geq 1}{2w_1+w_2 = 1}}}^{\left\lfloor \frac{n}{g} \right\rfloor}\frac{g-1}{2} \geq 1 + {\displaystyle}\sum_{w_1 = 1 }^{\left\lfloor \frac{n}{2g} \right\rfloor}\frac{g-1}{2} = 1 + \frac{g-1}{2} \left\lfloor \frac{n}{2g} \right\rfloor,$$ as claimed.

Positivity for bar-core partitions.
-----------------------------------

We have the following bar-analogues of Theorem \[infinitelymanystcores\] and Theorem \[f\].

\[infinitelymanystbarcores\] Let $s',t',g$ be odd, nontrivial integers. Then there are infinitely many $(\overline{s'g},\overline{t'g})$-cores which are not ${\bar{g}}$-cores.

There are infinitely many $\bar{g}$-cores for $g>1$. For any ${\bar{g}}$-core ${\gamma}$, consider the (completely and uniquely defined) bar-partition ${\lambda}$ with ${\mathcal Cor}_g({\lambda})={\gamma}$ and ${\mathcal Quo}_g({\lambda})=({\lambda}_0, \, {\lambda}_1, \, \ldots , \, {\lambda}_{\frac{g-1}{2}})=( (1), \, \emptyset, \, \ldots , \, 