\frac{n}{g} \right\rfloor} Q_{t'}(g,w)$, where $Q_{t'}(g,w)$ is the number of $g$-quotients of weight $w$ all of whose components are $t'$-cores.

In particular, $\hat{\psi}_{t \backslash g}(n) \geq g$. If, furthermore, $t' \geq 4$, then $\hat{\psi}_{t \backslash g}(n) \geq g \left\lfloor \frac{n}{g} \right\rfloor$.

First note that a partition ${\lambda}$ of $n$ is not a $g$-core if and only if it has $g$-weight $w \geq 1$. We therefore have (as in the proof of Theorem \[stcore\]) $$\hat{\psi}_{t \backslash g}(n)  =  {\displaystyle}\sum_{w \geq 1} Q_{t'}(g,w) f_g(n-wg)  =  {\displaystyle}\sum_{w = 1}^{\left\lfloor \frac{n}{g} \right\rfloor} Q_{t'}(g,w) f_g(n-wg) \geq  {\displaystyle}\sum_{w = 1}^{\left\lfloor \frac{n}{g} \right\rfloor} Q_{t'}(g,w),$$ the last inequality holding by Theorem \[Ono\] since $g \geq 4$.

Note that, for any $w \geq 1$, we have $Q_{t'}(g,w) \geq g f_{t'}(w)$, as one can get $g$ different $g$-tuples of $t'$-cores of weight $w$ by placing a $t'$-core of $w$ (if any exists) in any of the $g$ positions available. We therefore get $$\hat{\psi}_{t \backslash g}(n)  \geq {\displaystyle}\sum_{w = 1}^{\left\lfloor \frac{n}{g} \right\rfloor} g f_{t'}(w).$$ Now, since $t'>1$, the partition $(1)$ is the only $t'$-core of 1, whence $f_{t'}(1)=1$ (and, in fact, $Q_{t'}(g,1)=g$), so that $\hat{\psi}_{t \backslash g}(n)  \geq g f_{t'}(1)=g$, as claimed.

If, furthermore, $t' \geq 4$, then, by Theorem \[Ono\], $f_{t'}(w) 