\, \ldots , \, {\lambda}_{\frac{t-1}{2}}$.

Now consider odd integers $s=s'g$ and $t=t'g$, where $g>1$. We denote by $\xi_{{\bar{s}},\bar{t}}$ the generating function for the number of $({\bar{s}},\bar{t})$-core partitions. We now have the following bar-analogues of Theorem \[stcore\] and Corollary \[genfunstcore\].

\[stbarcore\] Let ${\lambda}$ be any bar-partition, and let ${\mathcal Quo}_{\bar{g}}({\lambda})=({\lambda}_0, \, {\lambda}_1, \, \ldots , \, {\lambda}_{\frac{g-1}{2}})$ be the ${\bar{g}}$-quotient of ${\lambda}$. Then ${\lambda}$ is an $({\bar{s}},\bar{t})$-core if and only if

1.  ${\lambda}_0$ is an $(\bar{s}', \bar{t}')$-core and

2.  ${\lambda}_1, \, \ldots , \, {\lambda}_{\frac{g-1}{2}}$ are $(s',t')$-cores.

Note that ${\lambda}$ is an ${\bar{s}}$-core if and only if ${\lambda}$ has no bar of length divisible by $s=s'g$. Now, by Proposition \[barbij\], for any $k \geq 1$, there is a bijection between the set of bars of length $ks=kgs'$ in ${\lambda}$ and the set of bars of length $ks'$ in ${{\mathcal Quo}}_{\bar{g}}({\lambda})$. Hence ${\lambda}$ is an ${\bar{s}}$-core if and only if ${\lambda}_0$ is an $\bar{s'}$-core and ${\lambda}_1, \, \ldots , \, {\lambda}_{\frac{g-1}{2}}$ are $s'$-cores. Similarly, ${\lambda}$ is a $\bar{t}$-core if and only if ${\lambda}_0$ is a $\bar{t'}$-core and ${\lambda}_1, \, \ldots , \, {\lambda}_{\frac{g-1}{2}}$ are $t'$-cores. The result follows.

\[genfunstbarcore\] With the above notation, we have $$\xi_{{\bar{s}},\bar{t}}(x)= \xi_{\bar{s}',\bar{t}'}(x^g)  \psi_{s',t'}(x^g)^{\frac{g-1}{2}}  F_{\bar{g}}(x),$$ where $F_{{\bar{g}}}(x)$ is 