Consider $V\_{(n-1, 1)}$, the $n-1$ dimensional irreducible representation of $S\_n$, i.e. the "standard" or "defining" representation. Is there a nice formula for how the $k$-th tensor power of $V\_{(n-1, 1)}$ decomposes into irreps?



Tensor powers of the standard representation
For convenience consider the representation $Y=V\_n\oplus V\_{n-1,1}$ instead of $V\_{n-1,1}$. Then the multiplicity of the representation of $S\_n$ indexed by the partition $\lambda$ of $n$ in the $k$th tensor power of $Y$ equals the scalar product of the symmetric function $s\_1^k$ (where $s\_1=x\_1+x\_2+\cdots$ denotes a Schur function) with the plethysm $s\_\lambda[1+h\_1+h\_2+h\_3+\cdots]$, where $h\_i$ is the complete symmetric function of degree $i$. This follows from the theory of inner plethysm; see Exercise 7.74 of *Enumerative Combinatorics*, volume 2. Since plethysm is in general intractable, I don't expect anything much simpler. This result does allow, however, these decompositions to be computed using Stembridge's Maple package SF for small values of $n$ and $k$.


**Addendum.** I used the method of Exercise 7.74 to get the analogous result for $V\_{n-1,1}$. 
Namely, the multiplicity of the representation of $S\_n$ indexed by the partition $\lambda$ of $n$ in the $k$th tensor power of $V\_{n-1,1}$ equals the scalar product of $s\_1^k$ with the symmetric function $(1-e\_1+e\_2-e\_3+\cdots)\cdot s\_\lambda[1+h\_1+h\_2+h\_3+\cdots]$, where $e\_i$ is an elementary symmetric function.


**Addendum #2.** A alternative formulation is the following. The multiplicity of the representation of $S\_n$ indexed by the partition $\lambda$ of $n$ in the $k$th tensor power of $V\_{n-1,1}$ equals the scalar product of $(s\_1-1)^k$ with the symmetric function $s\_\lambda[1+h\_1+h\_2+h\_3+\cdots]$.


**News flash!** I said above that plethysm in in general intractable. Indeed, the Schur function expansion of $s\_\lambda[1+h\_1+h\_2+\cdots]$ looks hopeless to me. However, taking the scalar product with $s\_1^k$ results in a lot of simplification. I can show the following. The multiplicity of the representation of $S\_n$ indexed by the partition $\lambda$ of $n$ in the $k$th tensor power of $V\_n\oplus V\_{n-1,1}$ equals the coefficient of $s\_\lambda$ in the Schur function expansion of $(1+h\_1+h\_2+\cdots)\cdot \sum\_{j=1}^k
S(k,j)s\_1^j$, where $S(k,j)$ is a Stirling number of the second kind. (After obtaining this result, I noticed that it is essentially the same as Corollary 2 of the Goupil-Chauve paper mentioned in Vasu Vineet's comment.) Since for fixed $j$ we have $S(k,j)=\frac{1}{j!}\sum\_{i=1}^j (-1)^{j-i}{j\choose i}i^k$, we can get explicit formulas for the multiplicities for fixed $\lambda$ that don't involve Stirling numbers. For instance, when $\lambda=(3)$ the multiplicity is $\frac{1}{6}(3^k+3)$, for $\lambda=(2,1)$ it is $3^{k-1}$, and for $\lambda=(1,1,1)$ it is $\frac{1}{6}(3^k-3)$. In particular, the multiplicity for $\lambda = (1^n)$ (i.e., $n$ parts equal to 1) is $S(k,n)+S(k,n-1)$.


