self-conjugate $(s,t)$-core partitions of $n$.

If ${\lambda}$ is any self-conjugate $(s,t)$-core of $n$ of $g$-weight $v$, then ${\mathcal Cor}_g(\lambda)$ can be any self-conjugate $g$-core of $n-vg$. If $g$ is even, then ${\mathcal Quo}_g(\lambda)$ has the form $({\lambda}_0, \ldots , {\lambda}_{\frac{g}{2}-1}, {\lambda}^{\vee}_{\frac{g}{2}-1}, \ldots , {\lambda}^{\vee}_0)$, and $({\lambda}_0, \ldots , {\lambda}_{\frac{g}{2}-1})$ can be any $\frac{g}{2}$-tuple of $(s',t')$-cores whose sizes add up to $w$ such that $v=2w$. The number of such ${\lambda}$’s is thus $$\hat{\psi}^*_{s,t}(n)={\displaystyle}\sum_{w\geq 0} Q_{s',t'}\left( \frac{g}{2},w \right) f^*_g(n-2wg)$$ in this case.

If $g$ is odd, then $${\mathcal Quo}_g(\lambda)=({\lambda}_0, \ldots , {\lambda}_{\frac{g-3}{2}}, {\lambda}_{\frac{g-1}{2}}, {\lambda}^{\vee}_{\frac{g-3}{2}}, \ldots , {\lambda}^{\vee}_0)$$ where $({\lambda}_0, \ldots , {\lambda}_{\frac{g-3}{2}})$ can be any $\frac{g-1}{2}$-tuple of $(s',t')$-cores whose sizes add up to some $w_1$, and ${\lambda}_{\frac{g-1}{2}}$ can be any self-conjugate $(s',t')$-core of $w_2$, with $v=2w_1+w_2$. The number of such ${\lambda}$’s is thus $$\hat{\psi}^*_{s,t}(n)= {\displaystyle}\sum_{2w_1+w_2\geq 0} Q_{s',t'}\left( \frac{g-1}{2},w_1 \right) f^*_{s',t'}(w_2) f^*_g(n-(2w_1+w_2)g)$$ in this case. These yield the desired generating functions.

Simultaneous bar-cores with common divisor.
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In this section, we state and prove new bar-analogues of Theorem \[stcore\] and Corollary \[genfunstcore\].

We start with combinatorial properties of bar-partitions. For more details, we refer the reader to [@O3 Section 4]. A $\bar{t}$-core, or $\bar{t}$-core partition, is any bar-partition which has no bar of 