Decomposing Sets of Inversions

In this note we consider the question how the set of inversions of a permutation $π\in S_n$ 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 $π$ and the modular decomposition of its inversion graph. We also consider the special case of multiplicative decompositions.

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