Semistandard Young tableau

A semistandard Young tableau fills a Young diagram with positive integers, permitting repeated entries, so that entries weakly increase along rows and strictly increase down columns. Its content records the number of occurrences of each entry. The Kostka number $K_{\lambda \mu }$ counts such fillings of shape $\lambda$ and content $\mu$.

For adjacent letters $i,i+1$, pair entries in the same column and leave those pairs fixed. In each row, interchange the numbers of unpaired $i$ and $i+1$ entries, preserving their weak order. Strict columns are preserved: a letter that could obstruct a change from $i$ to $i+1$, or conversely, would already have been paired. Applying the operation twice restores the tableau. The paired letters contribute equal amounts to the two contents, so the involution swaps their total multiplicities.

A word in positive integers is a lattice word if every initial segment has at least as many $i$'s as $(i+1)$'s for every $i$. This is the reading-word condition in the Littlewood–Richardson rule. Under good-letter matching in a tableau word, every letter of a lattice word is good.