partition.\] A [*hook*]{} ${\mathcal H}_{\iota \gamma}$ of $[\lambda]$ with [*corner*]{} $(\iota, \gamma)$, using matrix notation, is the set of boxes in the same row and to the right of $(\iota, \gamma)$, in the same column and below $(\iota, \gamma)$, and $(\iota, \gamma)$ itself. Then ${\mathcal H}=\{{\mathcal H}_{\iota,\gamma} \, | \, \iota, \gamma \geq 1\}$ is the [*hook set*]{} of $\lambda$. The set $\{{\mathcal H}_{\iota1} \, | \, \iota \geq 1\}$ consists of the [*first column hooks*]{} of \[$\lambda$\]. The [*hook length*]{} $h_{\iota \gamma}$ (or (or $|{\mathcal H}_{\iota\gamma}|$) of ${\mathcal H}_{\iota \gamma}$ is the number of boxes in the hook. The set $\{h_{\iota \gamma}\}$ will be called the [*set of hook lengths*]{} of $\lambda$. In particular, the set $\{h_{\gamma}\}$ of [*first column hook lengths*]{} of $\lambda$ completely determine the partition. The Young diagram $[\lambda^{\vee}]$ of the [*conjugate partition*]{} $\lambda^{\vee}$ of $\lambda$ is obtained by exchanging rows and columns of the Young diagram of $[\lambda]$. We say $\lambda$ is a [*self-conjugate*]{} partition if $\lambda=\lambda^{\vee}$. A self-conjugate partition is completely determined by its [*set of diagonal hook lengths*]{} $\Delta(\lambda)=\{h_{ii}\}$.

Let $\{h_{\iota\gamma} \, | \, \iota, \gamma \geq 1\}$ be the [*multiset of hook lengths of*]{} $\lambda$ (including repetitions). A [*t-hook*]{} is a hook 