$(\bar{s},\bar{t})$-core partitions when $s,t,g$ are all odd and nontrivial.

In Section 5 we show how the generating functions mentioned above inherit Ramanujan-type congruences.

Generating functions.
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Simultaneous core partitions with common divisor
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We start with some relevant background. For more details we refer the reader to [@O3 Section 3]. A [*t-core*]{} or [*t-core partition*]{} is any partition whose set of hook lengths avoids $t$, or has no $t$-hooks. More generally, the [*t-core*]{} of $\lambda$, denoted ${\mathcal Cor}_t{({\lambda})}$ is obtained by removing from ${\lambda}$ all of its hooks of length divisible by $t$. If a partition has $w$ such $t$-hooks, then the [*t-quotient*]{} ${\mathcal Quo}_t({\lambda})$ is an $t$-tuple $({\lambda}_1, \, \ldots , \, {\lambda}_t)$ of partitions whose sizes add up to $w$. The integer $w$ is then called the [*$t$-weight*]{} of ${\lambda}$, and ${\mathcal Cor}_t({\lambda})$ is a partition of $n-wt$. The fundamental property of ${\mathcal Quo}_t({\lambda})$ is that it contains all the information about the hooks of length divisible by $t$ in ${\lambda}.$ The following lemma and proposition are well-known.

If $\lambda$ is a partition of $n$, and if $t$ is any positive integer, then $\lambda$ is completely and uniquely determined by its [*$t$-core*]{} ${\mathcal Cor}_t{({\lambda})}$ and [*$t$-quotient*]{} ${\mathcal Quo}_t({\lambda})$.

