We study the structure and enumeration of the final two 2x4 permutation classes, completing a research program that has spanned almost two decades. For both classes, careful structural analysis produces a complicated functional equation. One of these equations is solved with the guess-and-check paradigm, while the other is solved with kernel method-like techniques and Gröbner basis calculations.
We thank Michael Albert for his open-source software PermLab, which was a useful aide in this research.