 Solution

For shape and type both $\mu$, column strictness makes every entry in row $r$ at least $r$. All ${\mu }_1$ entries equal to one must therefore be in the first row, which has exactly that length; it is filled with ones. Remove that row and repeat after subtracting one from the remaining labels. Induction gives the unique filling whose $i$th row consists of $i$'s. Thus $K_{\mu \mu }=1$, and Young's rule proves that $S^{\mu }$ occurs exactly once.

More generally, if a semistandard filling of shape $\lambda$ and type $\mu$ exists, its entries at most $k$ must lie in the first $k$ rows. Counting them givesThese are the inequalities defining the dominance order on partitions. Hence $K_{\lambda \mu }>0$ implies $\boxed{\lambda \ge \mu }$, and the asserted restriction on summands follows from Young's rule.