Factoring Pattern-Free Permutations into Separable ones

TL;DR

This paper proves that permutations avoiding a specific pattern can be decomposed into a bounded number of separable permutations, with a linear-time algorithm, leveraging the concept of permutation width related to twin-width.

Contribution

It introduces a linear-time factorization method for pattern-avoiding permutations into separable permutations, based on the notion of permutation width and twin-width.

Findings

Permutations avoiding a pattern can be expressed as a product of bounded separable permutations.

The factorization can be computed efficiently in linear time for any fixed pattern.

Applications include graph classes of bounded twin-width and their first-order transductions.

Abstract

We show that for any permutation $\pi$ there exists an integer $k_{\pi}$ such that every permutation avoiding $\pi$ as a pattern is a product of at most $k_{\pi}$ separable permutations. In other words, every strict class $\mathcal C$ of permutations is contained in a bounded power of the class of separable permutations. This factorisation can be computed in linear time, for any fixed $\pi$. The central tool for our result is a notion of width of permutations, introduced by Guillemot and Marx [SODA '14] to efficiently detect patterns, and later generalised to graphs and matrices under the name of twin-width. Specifically, our factorisation is inspired by the decomposition used in the recent result that graphs with bounded twin-width are polynomially $\chi$-bounded. As an application, we show that there is a fixed class $\mathcal C$ of graphs of bounded twin-width such that every class of bounded twin-width is a first-order transduction of $\mathcal C$.