Decomposing Sets of Inversions

In this note we consider the question how the set of inversions of a permutation $\pi \in S_n$ can be partitioned into two subset, such that those are itself inversion sets of permutations. This is archived by exploiting a connection to a graph theoretical result. For this we establish a correspondence between the substitution decomposition of $\pi$ and the modular decomposition of its inversion graph. We also consider the special case of multiplicative decompositions.