I do not have the book at hand to check, but maybe you do...

Loday defines in his book on Cyclic homology what a crossed simplicial group is and for each such gadget a corresponding homology theory. There is, for example, the "cyclic" crossed simplicial group, and the corresponding homology theory is cyclic homology. Well, there is a crossed simplicial groups---call it $\Delta S$---built from symmetric groups, and it turns out that the corresponding homology theory coincides with Hochschild homology. I am pretty sure your complex is the analogue of Connes' $\lambda$-complex corresponding to the homology theory for $\Delta S$, which is quasi-isomorphic to the complex which defines the actual $\Delta S$-homology rationally and, therefore, should just give you simplicial homology over $\mathbb Q$.

Integrally, you should not take invariants in each degree, but consider a double complex, each of whose columns are the complex which computes $H^\bullet(S_{n+1},\mathord-)$, much as the double complex which computes cyclic homology has columns which compute $H^\bullet(C_{n+1},\mathord-)$.

I do not have the book at hand to check, but maybe you do...

Loday defines in his book on Cyclic homology what a crossed simplicial group is and for each such gadget a corresponding homology theory. There is, for example, the "cyclic" crossed simplicial group, and the corresponding homology theory is cyclic homology. Well, there is a crossed simplicial groups---call it $\Delta S$---built from symmetric groups, and it turns out that the corresponding homology theory coincides with Hochschild homology. I am pretty sure your complex is the analogue of Connes' $\lambda$-complex corresponding to the homology theory for $\Delta S$, which is quasi-isomorphic to the complex which defines the actual $\Delta S$-homology rationally and, therefore, should just give you simplicial homology over $\mathbb Q$.

Integrally, you should not take invariants in each degree, but consider a double complex, each of whose columns are the complex which computes $H^\bullet(S_{n+1},\mathord-)$, much as the double complex which computes cyclic homology has columns which compute $H^\bullet(C_{n+1},\mathord-)$.

(I would love to have an example where taking invariants integrally does not work, by the way!)

I do not have the book at hand to check, but maybe you do...

Loday defines in his book on Cyclic homology what a crossed simplicial groups and defines for each such object a corresponding homology theory. There is a classification of crossed simplicial groups: for example, one of them is the cyclic simplicial group, and the corresponding homology theory is cyclic homology. Well, one of the crossed simplicial groups---call it $\Delta S$---is built from symmetric groups, and it turns out that the corresponding homology theory coincides with Hochschild homology. I am pretty sure your complex is the analogue of Connes' $\lambda$-complex corresponding to the homology theory for $\Delta S$, which is quasi-isomorphic to the complex which defines the actual $\Delta S$-homology rationally and, therefore, should just give you simplicial homology over $\mathbb Q$.

Integrally, you should not take invariants in each degree, but consider a double complex, each of whose columns are the complex which computes $H^\bullet(S_{n+1},\mathord-)$, just as the double complex which computes cyclic homology.

I do not have the book at hand to check, but maybe you do...

Loday defines in his book on Cyclic homology what a crossed simplicial group is and for each such gadget a corresponding homology theory. There is, for example, the "cyclic" crossed simplicial group, and the corresponding homology theory is cyclic homology. Well, there is a crossed simplicial groups---call it $\Delta S$---built from symmetric groups, and it turns out that the corresponding homology theory coincides with Hochschild homology. I am pretty sure your complex is the analogue of Connes' $\lambda$-complex corresponding to the homology theory for $\Delta S$, which is quasi-isomorphic to the complex which defines the actual $\Delta S$-homology rationally and, therefore, should just give you simplicial homology over $\mathbb Q$.

Integrally, you should not take invariants in each degree, but consider a double complex, each of whose columns are the complex which computes $H^\bullet(S_{n+1},\mathord-)$, much as the double complex which computes cyclic homology has columns which compute $H^\bullet(C_{n+1},\mathord-)$.