= Solution
The inclusion here is inclusion of <Young diagrams>, not the <dominance order on partitions>. The <skew Young diagram> $\lambda/\mu$ is the set difference of their cells. Two cells are adjacent when they share an edge. It is connected when any two cells can be joined by such steps, and is a <rim hook> when it is connected and contains no $2\times2$ square. Under the usual edge-adjacency convention a <totally disconnected skew Young diagram> has only singleton components, equivalently no two cells share an edge. A <horizontal strip> instead means at most one cell per column; this distinction matters for the last part of this question.
A <standard skew Young tableau> is a <linear extension of a partially ordered set>: the cells are ordered by the row and column inequalities, and the tableau lists them in increasing label order. Write the current list as $x_1,\ldots,x_k$ and let $r_j$ be the desired label of $x_j$ in $R$. Then $(r_1,\ldots,r_k)$ is the one-line notation of the unique permutation $\pi$ with $\pi T=R$.
Whenever this list of desired labels is not increasing, there is an adjacent descent $r_j>r_{j+1}$. The cells $x_j,x_{j+1}$ are incomparable in the cell <partial order>: if they were comparable, both $T$ and $R$ would have to put them in the same order. Interchanging their consecutive current labels is therefore admissible. It removes exactly one <inversion of a permutation> from the list of desired labels. Repeating ends at $R$ after exactly the original inversion count, which is the <Coxeter length> $\ell(\pi)$. Thus
$$
\boxed{T\longrightarrow R\text{ by exactly }\ell(\pi)\text{ admissible adjacent swaps}.}
$$
This proves the <reduced adjacent-swap path between linear extensions> and applies equally to ordinary tableaux.
Back to article page
