Let $V=\{x_1,\dots,x_{n-1}\}$ be a set and take $X_n$ the simplicial complex of all the faces of dimension $n-4$, i.e. all the subsets of $V$ with $n-3$ elements (And their subsets). Is there an elementary way to compute its homology over a given field $k$?
If not, are there any results about it?
For example, for $n=5$ we have $n-4=1$, so the faces are segments, $V=\{x_1,x_2,x_3,x_4\}$ and $X_n\cong K_4$ which has (reduced) homology $\tilde{H}_1(X;k)\cong k^3$ and $\tilde{H}_i(X;k)=0$ elsewhere. Similarly when $n=6$ we have $\tilde{H}_2(X_n)=k^4$ and $\tilde{H}_i(X;k)=0$ for $i\neq 2$.
I'm particularly interested in inductive arguments, but any kind of argument works.
So, $X_n$ is the $(n-4)$-skeleton of the $(n-2)$ simplex $\Delta_{n-2}$. Then is dimensions $k<n-4$, $X_n$ and $\Delta_{n-2}$ have the same homology, which is zero in the case of reduced homology. For $k>n-4$ then the homology of $X_n$ is also zero, since $X_n$ has no $k$-simplices.
The only interesting case is $H_{n-4}(X_n)$. This can be found using Euler characteristics; $$\sum_{j=0}^{n-4}(-1)^j\dim_k H(X_n,k)=\sum_{j=0}^{n-4}(-1)^j\binom{n-1}{j+1},$$ that is $$1+(-1)^{n-4}\dim_k H_{n-4}(X_n,k)=0+(-1)^n\binom{n-1}{n-1}-(-1)^n\binom{n-1}{n-2}$$ etc.