Permutation character of the symmetric group on subsets of certain size The symmetric group $S_n$ acts on $[n]:=\{1,\ldots,n\}$, thereby inducing an action on the set $$\wp_k(n)=\{\: A\subseteq[n] \::\: \#A=k \:\}$$ of subsets of cardinality $k$, simply by $$(g,A)\mapsto g(A)=\{\: g(a) \::\: a\in A \:\}.$$
This finally yields an action on $V_k:=\mathbb C[\wp_k(n)]$.
By Lemma 4 in this paper (arXiv:0903.2864), the character of $V_k$ is given as
$$\tag1 \sum_{r=0}^k \chi_{(n-r,r)}$$
where $\chi_\lambda$ is the irreducible character of $S_n$ corresponding to the partition $\lambda\vdash n$. There is no proof of Lemma 4 in the above reference, but the author says that the result is due to Frobenius. I would like to have a reference of $(1)$ - it doesn't have to be the original paper of Frobenius, in fact I would prefer a more recent work which also has a proof.
 A: A classical construction of the Specht modules of $S_n$ says that $\chi_{(n-k,k)}$ is present in the character $\pi_k$ of the action $S_n$ on $V_k$, with multiplicity 1. Indeed $M^\lambda:=V_k$ is the permutation module arising along the way of constructing the Specht module $S^\lambda$ for the partition $\lambda=(n-k,k)$. 
Moreover, it is easy to see that the centralizer of the action of $S_n$ on $V_k$ in the full matrix algebra of $\binom{n}{k}\times\binom{n}{k}$ is commutative, and has dimension $k+1$ (The centralizer is spanned as an algebra by the 0-1 matrices corresponding to the orbits of $S_n$ on the ordered pairs of $k$-subsets - this is a general fact about permutation representations of finite groups; here these matrices are symmetric, and thus it's a commutative algebra). Thus $\pi_k$ is a sum of $k+1$ irreducible characters, each of them with multiplicity 1. At this moment we know two of them, namely $\chi_{(n-k,k)}$ and $\chi_{(n)}$ (the latter is there, as it's the trivial character, present in every permutation character).
There is a description (see e.g. Volume 2 of the Richard Stanley's book) of irreducible characters arising in $M^\lambda$, for any $\lambda$. Namely, 
$$M^\lambda=S^\lambda\oplus \oplus_{\mu\triangleright\lambda} 
K_{\lambda\mu} S^\mu,$$
where $\triangleright$ stands for the dominance partial ordering on the partitions of $n$, and the $K_{\lambda\mu}$'s are famous Kostka numbers (in our case they are all 0 or 1). Using this, one can easily complete the proof of (1).

PS. In Stanley's book, this question is Example 7.18.8 on p.355, Volume 2.  
