On some generalization of the M\"obius configuration

The M\"obius $(8_4)$ configuration is generalized in a purely combinatorial approach. We consider $(2n_n)$ configurations ${\goth M}_{(n,\varphi)}$ depending on a permutation $\varphi$ in the symmetric group $S_n$. Classes of non-isomorphic configurations of this type are determined. The parametric characterization of ${\goth M}_{(n,\varphi)}$ is given. The uniqueness of the decomposition of ${\goth M}_{(n,\varphi)}$ into two mutually inscribed $n$-simplices is discussed. The automorphisms of ${\goth M}_{(n,\varphi)}$ are characterized for $n\geq 3$.