a partition of $n$ which is a $t'g$-core but not a $g$-core.

Aukerman, Kane and Sze do not consider $(\bar{s},\bar{t})$-core partitions with $g>1$; they are considered here for the first time.

Outline of paper
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We begin, in Section 2.1, by reproving, in a succinct fashion, Theorems 1.6 and 1.8 and Corollary 1.7 of Aukerman, Kane, and Sze using known properties of the $g$-core and $g$-quotient. We extend these results to self-conjugate $(s,t)$-core partitions and $(\bar{s},\bar{t})$-core partitions in Section 2.2 and Section 2.3.

In Section 3.1 we again reprove Corollary 1.9 in a succinct fashion before strengthening Theorem 1.10 for $g\geq 4$. In Section 3.2 and Section 3.3 we move to analogous results for self-conjugate $(s,t)$-core partitions and $(\bar{s},\bar{t})$-core partitions. In each case, we also set $s=t$ to strengthen existing positivity.

We give a new, shorter proof of Theorem \[Yangg\] in Section 4.1. In Section 4.2 we provide a bijection between self-conjugate $(s,t)$-core partitions and $(\bar{s},\bar{t})$-partitions when $s$ and $t$ are odd and $g=1$ using the work of Ford, Mai and Sze, Bessenrodt and Olsson. We combine those results with our work from Section 2.3 to show in Section 4.3 that there is a bijection between self-conjugate $(s,t)$-core partitions and 