Factoring Pattern-Free Permutations into Separable ones

We show that for any permutation π there exists an integer k π such that every permutation avoiding π as a pattern is a product of at most k π separable permutations. In other words, every strict class 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 π.

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 χ-bounded. As an application, we show that there is a fixed class C of graphs of bounded twin-width such that every class of bounded twin-width is a first-order transduction of C.