Skew Kostka coefficients from Littlewood-Richardson Coefficients I read on page 4 here 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.
 A: 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.
A: I believe most of what you want is in https://arxiv.org/abs/math/0308101 , especially the polynomiality you're looking for. Note that that was first proven in [H. Derksen, J. Weyman] "On the Littlewood-Richardson polynomials,"
http://www.math.lsa.umich.edu/~hderksen/preprints/lrpoly.dvi .
