On some generalization of the Möbius configuration

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

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