Skew Young diagram ($\lambda /\mu$)

For diagram inclusion $\mu \subseteq \lambda$, the skew Young diagram $\lambda /\mu$ consists of the cells of $\lambda$ outside $\mu$. Diagram inclusion is not the dominance order on partitions. Edge adjacency defines its connected components, and a connected skew diagram without a $2\times 2$ square is a rim hook.

A standard skew Young tableau fills the $k$ cells of a skew Young diagram bijectively with $1,\dots ,k$, increasing along each row and column. Equivalently it is a linear extension of a partially ordered set formed from the row and column inequalities. Interchanging consecutive labels preserves standardness exactly when their cells are incomparable in that partial order.

A horizontal strip is a skew Young diagram containing at most one cell in each column. Equivalently, $\lambda /\mu$ is a horizontal strip when ${\lambda }_i\ge {\mu }_i\ge {\lambda }_{i+1}$ for all $i$. The multiplicity of the trivial representation in the associated skew representation of a symmetric group is one for horizontal strips and zero otherwise. A column with two cells supplies a cyclic tableau vector on which an adjacent transposition acts by $-1$, excluding invariants.

Under edge adjacency, a skew Young diagram is totally disconnected when every component has one cell. This means that no two cells share an edge. It is stronger than being a horizontal strip, which excludes repeated columns but permits adjacent cells in one row. The trivial-constituent criterion for a skew representation of a symmetric group concerns horizontal strips, not total disconnection in this sense.