classes nevertheless share the same enumeration (a phenomenon called *Wilf-equivalence*), so the 2 $\times$ 4 classes have only $38$ different enumerations [@bona:the-permutation:; @kremer:permutations-wi:; @kremer:postscript:-per:; @kremer:finite-transiti:; @le:wilf-classes-of:]. This paper brings the number of $2\times 4$ Wilf classes which have been enumerated to $24$ (see Wikipedia [@wikipedia:enumerations-of:], which contains a list of such enumerations).

A central part of our approach depends on analysing the simple permutations in a class. An *interval* in the permutation $\pi$ is a set of contiguous indices $I=\{a,a+1,\dots,b\}$ such that the set $\{\pi(i){\::\:}i\in I\}$ is also contiguous. Every permutation $\pi$ of length $n$ has *trivial intervals* of lengths $0$, $1$, and $n$, and other intervals are called *proper*. A permutation with no proper intervals is called *simple*. Another way to think about simple permutations arises repeatedly throughout our arguments. Any subset $p_1,\dots$ of entries of the permutation $\pi$ defines a minimal axes-parallel rectangle (or simply, *box*), whose left edge slices through the leftmost of these entries, top edge slices through the greatest of these entries, and so on. A simple permutation is one in which the box defined by any proper subset of two or more of its entries is *separated* by an entry outside the box, by 