lengths of $\lambda$ are obtained by considering the hook lengths of the boxes in the shifted Young diagram embedded in the shift-symmetric diagram. $$\young(*86531,**431,***2,**,*)$$

We define a $t$[*-bar-core*]{}, or $\bar{t}$-core, to be a partition with no bar-hooks of length $t$. J. Baldwin, M. Depweg, B. Ford, A. Kumin, and L. Sze [@BDFS] proved a version of the positivity conjecture for self-conjugate core partitions; I. Kiming [@K2] proved a version of positivity for bar-core partitions. \[Kiming’s theorem was originally proven for $p$ prime (relevant for the so-called $p$-defect zero spin blocks), but the argument goes through without change for all odd $t\geq 7$ [@Ga]. We will use the more general statement.\]

\[fordsze\] Every non-negative integer $n$ has at least one self-conjugate $t$-core partition for $t=8$ or $t\geq 10$.

\[kimingtime\] Every non-negative integer $n$ has a $\bar{t}$-core partition for $t\geq 7$ and odd.

The next theorem is a recent result of J. Yang which connects self-conjugate $t$-core partitions and $\bar{t}$-core partitions.

[@Yang Theorem 1.1] \[Yangg\] Let $t$ be odd. There is a correspondence between the self-conjugate $t$-cores and $\bar{t}$-core partitions.

Generating functions
--------------------

Euler determined the generating function $P$ for the number $p(n)$ of partitions of $n$: $$P(x):=\sum^{\infty}_{n=0}p(n)x^n=\prod_{n=1}^\infty\frac{1}{1-x^n}$$ The generating function $F_t$ 