${\left\lfloor\frac{s}{2}\right\rfloor}\times {\left\lfloor\frac{t}{2}\right\rfloor}$ lattice which we call the [*diagonal hooks diagram*]{}. As a consequence, they find that, when $g=1$, the number of self-conjugate $(s,t)$-core partitions is $${\binom{{\left\lfloor\frac{s}{2}\right\rfloor}+{\left\lfloor\frac{t}{2}\right\rfloor}}{{\left\lfloor\frac{s}{2}\right\rfloor}}}.$$

A bar-partition is said to be an $({\bar{s}},\bar{t})$-core if it is both an ${\bar{s}}$-core and a $\bar{t}$-core. In the case that $s,t>1$ are odd and $g=1$, C. Bessenrodt and J. B. Olsson [@BO] construct a bijection between $(\bar{s},\bar{t})$-core partitions and monotonic paths in a $\frac{s-1}{2}\times\frac{t-1}{2}$ array, called the [*Yin-Yang diagram*]{}. As a consequence they find the number of $(\bar{s},\bar{t})$-core partitions and the size of the $(\bar{s},\bar{t})$-core which “contains" all others.

We outline the Ford, Mai and Sze construction as well as that of Bessenrodt and Olsson in Section 4.2.

Simultaneous core partitions with common divisor.
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When $g>1$, there are infinitely many $(s,t)$-cores, self-conjugate $(s,t)$-cores and $(\bar{s},\bar{t})$-cores, so there is no largest such partition. However, some properties of $(s,t)$-cores for $g=1$ remain true when $g>1$ (e.g. [@GN12] and [@Na07] generalize results in [@O4]). D. Aukerman, B. Kane and L. Sze first characterized an $(s,t)$-core partition ${\lambda}$ in terms of its $g$-core ${\mathcal Cor}_g({\lambda})$ and $g$-quotient ${\mathcal Quo}_g({\lambda})$, which we define formally in Section 2.1..

[@AKS Theorem 1.3]\[stcore\] Let ${\lambda}$ be any integer partition, $s=s'g$ 