Are the formulas for the multiplicity of $SL(w)$ invariants in $S^n(S^k(\mathbb{C}^w)$ known? This is a very classical topic. If no, in what ranges one can compute it (for certain paramaters fixed - e.g. it is obvious that this is zero for $n<w$)



Invariants in $S^n(S^k(\mathbb{C}^w)$
There is an isomorphism $S^d(\mathbb{C}^n) \cong V\_{n,d}$ where $V\_{n,d}$ is the the vector $\mathbb{C}$-space of $n$-ary forms of degree $d.$


The number $\nu\_{n,d}(k)$ of linearly independent homogeneous invariants of degree $k$ for $n$-ary form of degree $d$ is calculated by the formula:
$$
\nu\_{n,d}(k)=\sum\_{s \in \mathcal{W}} (-1)^{|s|} c\_{n,d}\bigl(k,(\rho-s(\rho))^\*\bigr). 
$$


here $c\_{n,d}(k,\mu) :=c\_{n,d}(k,(\mu\_1,\mu\_2,\ldots,\mu\_{n-1}))$ is the number of non-negative integer solutions of the system of equations
$$
\left \{
\begin{array}{l}
\omega\_1(\alpha)-\omega\_2(\alpha)=\mu\_1, \\
\ldots \\
\omega\_{n-2}(\alpha)-\omega\_{n-1}(\alpha)=\mu\_{n-2},\\
 \omega\_1(\alpha)+\omega\_2(\alpha)+\cdots +2\,\omega\_{n-1}(\alpha)=k\, d-\mu\_{n-1}, \\
|\alpha|=k,
\end{array}
\right.
$$
 $\rho$ is half the sum of the positive roots of Lie algebra $\mathfrak{sl\_{n}},$ $(-1)^{|s|}$ is the sign of the element $s \in \mathcal{W},$ $\mu^\*$ the unique dominant weight on the orbit $\mathcal{W}(\mu)$ of the Weyl group $\mathcal{W}$ of the Lie algebra $\mathfrak{sl\_{n}},$ and $\omega\_i(\alpha)$ is some functions defined on a $n$-ary form.


Details you may find in the paper


[L. Bedratyuk, Analogue of the Cayley-Sylvester formula and the Poincaré series for the algebra of invariants of n-ary form, Linear and Multilinear Algebra](https://docs.google.com/file/d/0B5kU1_wtaRBYTV9USFNZR0lOS0U/edit?pli=1)


