which we mean that this entry lies either

-   vertically amongst these entries but to the left (or right) of all of them (*vertical separation*), or

-   horizontally amongst these entries but above (or below) all of them (*horizontal separation*).

Simple permutations are precisely those that do not arise from a non-trivial inflation, in the following sense. Given a permutation $\sigma$ of length $m$ and nonempty permutations $\alpha_1,\dots,\alpha_m$, the *inflation* of $\sigma$ by $\alpha_1,\dots,\alpha_m$, denoted $\sigma[\alpha_1,\dots,\alpha_m]$, is the permutation of length $|\alpha_1|+\cdots+|\alpha_m|$ obtained by replacing each entry $\sigma(i)$ by an interval that is order isomorphic to $\alpha_i$ in such a way that the intervals are order isomorphic to $\sigma$. For example, $$2413[1,132,321,12]=4\ 798\ 321\ 56.$$ We give two particular types of inflations special terminology and notation. The inflation $12[\alpha_1,\alpha_2]$ is called a *(direct) sum* and denoted by $\alpha_1\oplus\alpha_2$. A permutation is *sum decomposable* if it can be expressed as a nontrivial sum, and *sum indecomposable* otherwise. The inflation $21[\alpha_1,\alpha_2]$ is called a *skew sum*, similarly denoted $\alpha_1\ominus\alpha_2$, and accompanied by analogous terms *skew decomposable* and *skew indecomposable*. We extend the notion of direct and skew sum to classes, defining $${\mathcal{C}}\oplus{\mathcal{D}}=\{\pi\oplus\sigma{\::\:}\pi\in{\mathcal{C}}\mbox{ and }\sigma\in{\mathcal{D}}\},$$ with an analogous definition for ${\mathcal{C}}\ominus{\mathcal{D}}$.

The precise 