Decomposing Sets of Inversions

TL;DR

This paper explores how to partition the inversion set of a permutation into two inversion sets, linking permutation decomposition to graph modular decomposition, and examines multiplicative cases.

Contribution

It establishes a novel connection between permutation inversion sets and graph modular decomposition, providing a new framework for understanding permutation structure.

Findings

Partitioning inversion sets relates to graph modular decomposition.

A correspondence between permutation substitution and graph modular decomposition is established.

Special cases of multiplicative decompositions are analyzed.

Abstract

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.