length divisible by $t$. More generally, ${{\mathcal Cor}}_{\bar{t}}({\lambda})$ the $\bar{t}$-core of ${\lambda}$ can be obtained by removing from ${\lambda}$ all of its bars of length divisible by $t$. If there are $w$ such $(t)$-bars, then the $\bar{t}$-quotient ${\mathcal Quo}_{\bar{t}}({\lambda})$ is an $\frac{t+1}{2}$-tuple $({\lambda}_0, \, {\lambda}_1, \, \ldots , \, {\lambda}_{\frac{t-1}{2}})$ of partitions whose sizes add up to $w$, where ${\lambda}_0$ is itself a bar-partition. The integer $w$ is then called the [*$\bar{t}$-weight*]{} of ${\lambda}$, and ${\mathcal Cor}_{\bar{t}}({\lambda})$ is a bar-partition of $n-wt$. The $\bar{t}$-quotient ${\mathcal Quo}_{\bar{t}}({\lambda})$ is said to have $t$-weight $w$.

[@O3 Proposition (4.2)] If $\lambda$ is a bar-partition, and if $t$ is any odd positive integer, then $\lambda$ is completely and uniquely determined by ${\mathcal Cor}_{\bar{t}}(\lambda)$ and ${\mathcal Quo}_{\bar{t}}({\lambda})$.

In analogy with Proposition \[hookbij\], we have the following:

\[barbij\] [@O3 Theorem (4.3)] If $t$ is an odd positive integer and ${\lambda}$ is a bar-partition, then, for any integer $k \geq 1$, there is a canonical bijection between the set of bars of length $kt$ in ${\lambda}$ and the set of bars of length $k$ in ${\mathcal Quo}_{\bar{t}}({\lambda})$, where a bar in ${\mathcal Quo}_{\bar{t}}({\lambda})$ is defined to be a bar in ${\lambda}_0$, or a hook in any of ${\lambda}_1, 