and there exists a self-conjugate $t$-core of $n$ which is not a $g$-core. In addition, if $n \geq 2g$, then $\hat{\psi}_{t^*\backslash g}(n)\geq \frac{g}{2}$.

If, furthermore, $g$ is even and $t'\geq 4$, then $\hat{\psi}_{t^*\backslash g}(n)\geq \frac{g}{2} \left\lfloor \frac{n}{2g} \right\rfloor$. If $g$ is odd and $t'=8$ or $t' \geq 10$, then $\hat{\psi}_{t^*\backslash g}(n)\geq 1+ \frac{g-1}{2} \left\lfloor \frac{n}{2g} \right\rfloor$.

Again, note that a partition ${\lambda}$ of $n$ is not a $g$-core if and only if it has $g$-weight $v \geq 1$. Following the proof of Theorem \[genfunselfstcore\], we see that the number $\hat{\psi}_{t^*\backslash g}(n)$ of self-conjugate $t$-cores of $n$ which are not $g$ cores is given by $$\hat{\psi}_{t^*\backslash g}(n) = {\displaystyle}\sum_{w \geq 1} Q_{t'}\left(\frac{g}{2},w\right) f^*_g(n-2wg) \;  \; \mbox{if $g$ is even},$$ and $$\hat{\psi}_{t^*\backslash g}(n) = {\displaystyle}\sum_{2w_1+w_2\geq1} Q_{t'}\left(\frac{g-1}{2},w_1\right)f^*_{t'}(w_2) f^*_g(n-(2w_1+w_2)g) \; \; \mbox{if $g$ is odd}.$$ Since $g=8$ or $g \geq 10$, we get, by Theorem \[fordsze\], $$\hat{\psi}_{t^*\backslash g}(n) = {\displaystyle}\sum_{w = 1}^{\left\lfloor \frac{n}{2g} \right\rfloor} Q_{t'}\left(\frac{g}{2},w\right) f^*_g(n-2wg) \geq \sum_{w = 1}^{\left\lfloor \frac{n}{2g} \right\rfloor} Q_{t'}\left(\frac{g}{2},w\right)  \; \; \mbox{if $g$ is even},$$ while $$\hat{\psi}_{t^*\backslash g}(n) \geq {\displaystyle}\sum_{2w_1+w_2 = 1}^{\left\lfloor \frac{n}{g} \right\rfloor} Q_{t'}\left(\frac{g-1}{2},w_1\right)f^*_{t'}(w_2)  \;  \; \mbox{if $g$ is odd}.$$ Suppose first that $g$ is even. If $\left\lfloor \frac{n}{2g} \right\rfloor =0$, i.e. if $n < 2g$, then $\hat{\psi}_{t^*\backslash g}(n)=0$. 