Hook, arm, leg, and hook length of a box

Let $\lambda ⊢n$ with English Young diagram $[\lambda ]$ (Partitions, English diagrams, and conjugation) and let $x=(i,j)\in [\lambda ]$. The hook of $x$ is the set $H(x):=\{(i,j^{\mathrm{\prime }})\in [\lambda ]:j^{\mathrm{\prime }}\ge j\}\cup \{(i^{\mathrm{\prime }},j)\in [\lambda ]:i^{\mathrm{\prime }}\ge i\},$ the union of the boxes of $[\lambda ]$ weakly to the right of $x$ in row $i$ and weakly below $x$ in column $j$; the box $x$ itself belongs to both parts and is counted once. The arm $\mathrm{arm}(x)$ is the part in row $i$ strictly to the right of $x$, so $\mathrm{arm}(x)=\{(i,j^{\mathrm{\prime }}):j<j^{\mathrm{\prime }}\le {\lambda }_i\}$; the leg $\mathrm{leg}(x)$ is the part in column $j$ strictly below $x$, so $\mathrm{leg}(x)=\{(i^{\mathrm{\prime }},j):i<i^{\mathrm{\prime }}\le {\lambda }_j^{\mathrm{\prime }}\}$. The arm length is $a(x):={\lambda }_i-j$, the leg length is $\mathrm{\ell }(x):={\lambda }_j^{\mathrm{\prime }}-i$, and the hook length is

$h(x):=a(x)+\mathrm{\ell }(x)+1={\lambda }_i-j+{\lambda }_j^{\mathrm{\prime }}-i+1,$

so that $H(x)$ has exactly $h(x)$ boxes. Here ${\lambda }_j^{\mathrm{\prime }}$ is the number of rows of $[\lambda ]$ of length at least $j$, so the boxes of column $j$ below row $i$ are exactly the rows $i+1,\dots ,{\lambda }_j^{\mathrm{\prime }}$ and the arm has ${\lambda }_i-j$ boxes; the arm, the leg and the anchor $x$ are pairwise disjoint and exhaust $H(x)$, which gives the count.

A box is removable in the sense of Removable and addable nodes if and only if it is the last box of its row and of its column, i.e. if and only if $h(x)=1$: a row endpoint $(i,{\lambda }_i)$ is removable exactly when no box lies immediately below it, that is when ${\lambda }_i>{\lambda }_{i+1}$, and then $a(x)={\lambda }_i-{\lambda }_i=0$ and $\mathrm{\ell }(x)={\lambda }_{{\lambda }_i}^{\mathrm{\prime }}-i=0$; conversely $h(x)=1$ forces $a(x)=\mathrm{\ell }(x)=0$, so $x$ ends both its row and its column. The empty partition has no boxes.

Finally $P(\lambda ):={\prod }_{x\in [\lambda ]}h(x)$ denotes the hook product of $\lambda$, the empty product $P(\mathrm{\varnothing })=1$ being part of the convention. This fixes the off-by-one convention used by the whole page: the anchor box contributes $1$, the arm contributes ${\lambda }_i-j$ and the leg contributes ${\lambda }_j^{\mathrm{\prime }}-i$. No choice principle is used.