Combinatorial aspects of mixed arrangements

In this paper, we consider mixed arrangements which are composed of the hyperplanes (or subspaces) and the spheres. We research the posets of their intersection sets and calculate the Möbius functions of the mixed arrangements through the hyperplane (or subspace) arrangements' Möbius functions. Moreover, by the method of deletion and restriction, we calculate the recurse formulas of the triples of these mixed arrangements.