Removable border strips and their height

Let $\lambda ⊢n$ be a partition with English Young diagram $[\lambda ]$, so that rows are numbered downward and columns rightward, and let $\nu \subseteq \lambda$ be a subpartition, so that the skew diagram $[\lambda \mathrm{/}\nu ]=[\lambda ]\setminus [\nu ]$ is the set of nodes of $[\lambda ]$ outside $[\nu ]$ (Partitions, English diagrams, and conjugation, Skew diagrams and semistandard skew tableaux). Two nodes are side-adjacent when their coordinates differ by exactly one in exactly one coordinate.

A removable border strip of $\lambda$ (also called a skew hook or rim hook) is a nonempty set $\xi$ of nodes of $[\lambda ]$ such that

(i) $\xi$ is edge-connected: any two nodes of $\xi$ are joined by a chain of side-adjacent nodes of $\xi$;

(ii) $\xi$ contains no $2\times 2$ block: there are no nodes $(i,j),(i+1,j),(i,j+1),(i+1,j+1)$ all belonging to $\xi$;

(iii) $[\lambda ]\setminus \xi$ is again the Young diagram of a partition, say $[\lambda ]\setminus \xi =[\nu ]$ for a partition $\nu ⊢n-\mathrm{\mid }\xi \mathrm{\mid }$.

Equivalently, a removable border strip is a skew diagram $\lambda \mathrm{/}\nu$ that is connected and contains no $2\times 2$ block; since the complement of $[\lambda \mathrm{/}\nu ]$ in $[\lambda ]$ is $[\nu ]$, condition (iii) is then automatic, and (i) and (ii) say exactly that the skew diagram is connected and free of $2\times 2$ blocks. We write $\xi =\lambda \mathrm{/}\nu$ in that case. The size of $\xi$ is its number of nodes $\mathrm{\mid }\xi \mathrm{\mid }$; a removable border strip of size $r$ is a removable $r$-strip, and then $\nu ⊢n-r$.

The height (also leg length) of a nonempty connected border strip $\xi$ is $\mathrm{ht}(\xi ):=(\text{number of distinct rows of }[\lambda ]\text{ occupied by }\xi )-1.$

The counting identity. If a removable border strip $\xi$ has $r$ nodes and occupies $c$ distinct columns and $w$ distinct rows of $[\lambda ]$, then $w=r-c+1$, hence $\mathrm{ht}(\xi )=r-c\ge 0$. Indeed $[\lambda \mathrm{/}\nu ]$ is a connected, $2\times 2$-free skew diagram with complement a Young diagram. Every strip node $(p,q)$ lies on the rim: if $(p+1,q+1)$ belonged to $[\lambda ]$, it could not belong to $[\nu ]$, since that would force $(p,q)\in [\nu ]$; and the partition inequalities then put all four nodes of the corresponding $2\times 2$ block in the strip, a contradiction. The rim nodes in row $p$ have columns $\max (1,{\lambda }_{p+1}),\dots ,{\lambda }_p$, where missing row lengths are zero. Read each such interval from right to left, then move down at column ${\lambda }_{p+1}$ to the next row: these are exactly the rim nodes and their side-adjacencies. Thus the rim is a single path moving left or down, and its connected subsets are subpaths. The $r-1$ steps of $\xi$ consist of $w-1$ downward steps and $c-1$ leftward steps, since the path visits exactly the $w$ occupied rows and exactly the $c$ occupied columns, whence $r-1=(w-1)+(c-1)$.

In particular $\mathrm{ht}(\xi )=0$ exactly when $\xi$ is a horizontal strip, that is, occupies at most one node in each column, and in general $c=r-\mathrm{ht}(\xi )$ is the number of columns occupied. A removable border strip of size $1$ is exactly a removable node (Removable and addable nodes): a single node is edge-connected and contains no $2\times 2$ block, and its removal leaves a Young diagram precisely when the node is removable. No choice principle is used.

• Border strips are the pieces of the rim that can be peeled off. Since $[\lambda \mathrm{/}\nu ]$ has complement the Young diagram $[\nu ]$, its nodes form a contiguous stretch of the lower-right rim of $[\lambda ]$: reading the strip from its north-east end to its south-west end one moves only leftward or downward, and the strip is determined by the pair $(\nu ,\lambda )$. This is the sense in which removable border strips are the "skew hooks" of James §18 and the "border strips" of Macdonald I §3 and §7.

• Height counts rows, not nodes in a column. A vertical strip, occupying a single column of $r$ nodes, has $c=1$ and $\mathrm{ht}=r-1$, the largest possible height for a strip of size $r$; the rows-occupied count $w=\mathrm{ht}+1=r$ then agrees with the node count. The height sign $(-1{)}^{\mathrm{ht}}$ is the one used in the Murnaghan–Nakayama rule (The Murnaghan–Nakayama character rule).

• Conventions. The empty set is not a removable border strip: condition (i) applies to a nonempty set, and the height of a strip requires at least one occupied row. A removable border strip is thus never empty, and the skew diagram $\lambda \mathrm{/}\lambda$ is valid notation for the empty diagram, but is not a removable border strip. The height convention agrees with the leg length $\mathrm{\ell }(x)$ of the hook of $x$ under the hook/rim-hook correspondence (Hooks of a partition correspond bijectively to removable border strips).