I read on page 4 [here](http://lipn.univ-paris13.fr/~toumazet/biblio/ARTICLES/NTB.pdf) that the Kostka coefficients $K\_{\lambda,\mu}$ are specializations of the Littlewood-Richardson coefficients $c^\tau\_{\sigma,\lambda}$ by specializing $\sigma,\tau$ depending on $\mu$ in a simple manner (certain sums of parts of $\mu$).


Is there a similar specialization/translation for Kostka coefficients obtained from skew shapes, $K\_{\lambda,\mu}^\nu$ where $\lambda/\nu$ is a skew shape?


Motivation: I would like to see if polynomiality of the map
$n \mapsto K\_{n \lambda, nw}^{n \nu}$ implies polynomiality for a similar map with LW-coefficients.



Skew Kostka coefficients from Littlewood-Richardson Coefficients
I prefer to write $K\_{\lambda/\nu,\mu}$ for $K^\nu\_{\lambda,\mu}$.
Using standard symmetric function notation, we have
 $$ K\_{\lambda/\nu,\mu}=\langle s\_{\lambda/\nu},h\_\mu\rangle =
 \langle s\_\lambda,s\_\nu h\_\mu\rangle. $$
Let $\rho/\sigma$ be a skew shape which is a disjoint union of shapes
$\nu, (\mu\_1), (\mu\_2), \dots$. Here $(\mu\_i)$ is a single row of
length $\mu\_i$. By "disjoint union," I mean that none of the shapes
has a square in the same row or in the same column as a square of
another of the shapes. Thus $s\_{\rho/\sigma} = s\_\nu h\_\mu$, so
 $$ K\_{\lambda/\nu,\mu}=\langle s\_\lambda,s\_{\rho/\sigma}\rangle =
 \langle s\_\lambda s\_\sigma, s\_\rho\rangle, $$
an ordinary Littlewood-Richardson coefficient.


