  Given partitions $\lambda,\mu,\nu$ with at most $n$ nonzero parts and a
permutation $w \in S_n$, the refined Littlewood-Richardson coefficient
$c_{\lambda \mu}^\nu(w)$ is the multiplicity of the irreducible $GL_n
\mathbb(C)$ module $V(\nu)$ in the so-called Kostant-Kumar submodule
$K(\lambda, w, \mu)$ of the tensor product $V(\lambda) \otimes V(\mu)$. We
derive a hive model for these coefficients and prove that the saturation
property holds if $w$ is $312$-avoiding, $231$-avoiding or a commuting product
of such elements. This generalizes the classical Knutson-Tao saturation
theorem.
