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 $\mathcal C$.

翻译：我们证明，对于任意一个排列π，存在一个整数k_π，使得每个避免模式π的排列都可以分解为至多k_π个可分离排列的乘积。换言之，每个严格的排列类C都包含在可分离排列类的一个有界幂次中。对于任意固定的π，该分解可在线性时间内计算得出。该结果的核心工具是由Guillemot和Marx [SODA '14] 引入的排列宽度概念（最初用于高效检测模式，后以双胞胎宽度之名推广至图与矩阵）。具体而言，该分解的构造灵感来源于近期关于有界双胞胎宽度的图具有多项式χ-有界性的研究成果。作为应用，我们证明存在一个具有有界双胞胎宽度的固定图类C，使得每个有界双胞胎宽度的图类都是C的一阶转导。