The fact that Kostka numbers equals to Littlewood-Richardson coefficients for some partitions is already known $\colon$ \begin{align} K_{\lambda \mu} = c_{\sigma \lambda}^\tau \end{align} where $\tau_i = \mu_{i} + \mu_{i+1} +\cdots, \sigma_i = \mu_{i+1} + \mu_{i+2} + \cdots$.

For example, it is written in Here, page 3.

But I have not found a proof of this claim. So if you know paper or book in which this claim's proof is written, then please tell me it. If possible I would like to know a proof which construct a bijection between semi-standard tableaux and LR-tableaux.