We investigate the structure of the two permutation classes defined by the sets of forbidden patterns \{{1234, 2341}\} and \{{1243, 2314}\}. By considering how the Hasse graphs of permutations in these classes can be built from a sequence of rooted source graphs, we determine their algebraic generating functions. Our approach is similar to that of “adding a slice”, used previously to enumerate various classes of polyominoes and other combinatorial structures. To solve the relevant functional equations, we make extensive use of the kernel method
