of length $t$. For fixed $t$, the partitions whose set of hook lengths does not contain any multiple of $t$ are known as [*t-core partitions*]{}. The [*positivity*]{} of $t$-core partitions (see Theorem \[Ono\] below) was proved in several steps: K. Erdmann and G. Michler [@EM] considered $t=5$ and $7$, K. Ono [@On] $4\leq t\leq11$, and A. Granville and K. Ono [@OG] completed the remaining cases. Later I. Kiming [@KO] shortened some arguments.

\[Ono\] Every non-negative integer $n$ has at least one $t$-core partition for $t\geq 4.$

[*Bar partitions*]{} are partitions with distinct parts. One obtains the [*shifted Young diagram*]{} of a bar partition by shifting the $i$th row in the Young diagram $i-1$ boxes to the right. The [*shift-symmetric diagram*]{} is then obtained by adjoining the parts of $\lambda$ as columns to the shifted diagram. Then there is an analogous definition of a [*bar-hook*]{} and [*bar-hook-length*]{}, seen on the shift-symmetric Young diagram. The bar-hook lengths are the hook lengths of the boxes of the shifted Young diagram embedded in the shift-symmetric Young diagram [@Be].

Consider the Young diagram of the bar partition $\lambda=(5,3,1)$ . $$\yng(5,3,1)$$ The shifted Young diagram is seen below, with bar-hook lengths included. $$\young(86531,:431,::2)$$ The bar hook 