for the number $f_t(n)$ of $t$-core partitions of $n$ was first obtained by J. B. Olsson [@O0]. $$F_{{t}}(x)=\sum^{\infty}_{n=0}f_t(n)x^n=\prod^{\infty}_{n=1}\frac{(1-x^{tn})^t}{(1-x^n)}.$$ F. Garvan, D. Kim, and D. Stanton (Eqs. 7.1(a) and (b) in [@GKS]) first described the generating function $F^*_t$ for the number $f^*_t(n)$ of self-conjugate $t$-core partitions of $n$. We use a formulation that appears in [@BDFS]. $$F^*_{t}(x)=\sum^{\infty}_{n=0}f^*_t(n)x^n=\left\{
\begin{array}{ll}
      \prod^{\infty}_{n=1}(1-x^{2tn})^{\frac{t}{2}}(1+x^{2n-1}) & \mbox{if $t$ is even,}\\\\
      \prod^{\infty}_{n=1}(1-x^{2tn})^{\frac{t-1}{2}}(\frac{1+x^{2n-1}}{1+x^{t(2n-1)}}) & \mbox{if $t$ is odd.} \\
\end{array} 
\right.$$ The generating function $F_{\bar{t}}$ for the number $f_{\bar{t}}(n)$ of $\bar{t}$-core partitions of $n$ was also found by Olsson [@O3]. $$F_{\bar{t}}(x)=\sum^{\infty}_{n=0}f_{\bar{t}}(n)x^n={\displaystyle}\prod_{n=1}^{\infty}\frac{(1-x^{2n})(1-x^{tn})^{\frac{t+1}{2}}}{(1-x^n)(1-x^{2tn})}$$

Simultaneous core partitions with no common divisor.
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A [*simultaneous core partition*]{} is a partition whose set of hook lengths avoids a specified subset of positive integers. An $(s,t)$-core partition is both an $s$-core and a $t$-core partition, where $s,t>1$. For the rest of the paper we let $g:=$gcd${(s,t)}$.

The following result by J. Anderson (Section 3, [@An]) describes, when $g=1$, the possible first-column hook lengths of an $(s,t)$-core partition.

\[andy\] Let $s,t>1$ and $g=1$. The first-column hook lengths of an $(s,t)$-core partition are of the form $(st-s-t)-(ks-lt)$ for some $k\geq 0$ and $l\geq 0$. In particular $st-s-t$ is the largest possible first-column hook length of an $(s,t)$-core 