hook lengths of $\lambda_{\pi}$ some $(s,t)$-core partition.

Let $s=7$ and $t=11$. Anderson’s lattice is shown below. The monotonic path $\pi$ appearing in the lattice corresponds to the $(7,11)$-core partition $\lambda_{\pi}$ whose first-column hook lengths $\{h_{\gamma1}\}_{1\leq \gamma\leq 5}=\{13,10,9,8,6,5,3,2,1\}$ are the values trapped between the (solid) path and the (dashed) $\pm$border. This corresponds to the partition $\lambda_{\pi}=(5,3^3,2^2,1^3).$

The condition on the monotonic paths above is equivalent to “staying above the diagonal" (also illustrated in Example 1.7). Anderson uses this to prove that the number of $(s,t)$-core partitions is $\frac{1}{(s+t)}\binom{s+t}{t}$. J. B. Olsson and D. Stanton [@OS] showed that the largest size of such an $(s,t)$-core partition is $\frac{(s^2-1)(t^2-1)}{24}$, verifying a conjecture of B. Kane [@Bk]. In particular such a largest size $(s,t)$-core is self-conjugate. The Armstrong conjecture [@AHJ], verified by P. Johnson [@Jo] and V. Wang [@Wa] (using different methods) says that the average size of a $(s,t)$-core partition is $\frac{(s-1)(t-1)(s+t+1)}{24}$ when $g=1$.

Simultaneous self-conjugate core and bar-cores with no common divisor.
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Let $s,t>1$ be odd. We use the notation $(s,t)^*$-core partition to indicate an $(s,t)$-core partition that is also self-conjugate. B. Ford, H. Mai and L. Sze [@fms] construct a bijection between $(s,t)^*$-core partitions and monotonic paths in a certain 