Skew diagrams and semistandard skew tableaux

Use the English row and column coordinates of Partitions, English diagrams, and conjugation. For partitions $\mu ,\lambda$ with $[\mu ]\subseteq [\lambda ]$, the skew diagram $[\lambda \mathrm{/}\mu ]$ is the set difference $[\lambda ]\setminus [\mu ]$.

A semistandard skew tableau of shape $\lambda \mathrm{/}\mu$ is a filling of these boxes by positive integers, weakly increasing from left to right in each row and strictly increasing from top to bottom in each column, using the same inequalities as Semistandard tableaux and Kostka numbers on the boxes that remain. Its weight is the finite sequence $\mathrm{wt}(T)=(a_1,a_2,\dots )$, where $a_r$ is the number of entries equal to $r$; its monomial is $x^{\mathrm{wt}(T)}={\prod }_{r\ge 1}{x}_r^{a_r}$.

A horizontal strip is a skew diagram with at most one box in each column. Two boxes are side-adjacent when their coordinates differ by one in exactly one coordinate. The edge-connected components of a skew diagram are the connected components of its boxes under side-adjacency. If $\mu =\lambda$, the skew diagram is empty and has exactly one filling, the empty tableau, of weight zero and monomial $1$.