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We investigate the structure of the two permutation classes defined by thesets of forbidden patterns {1234,2341} and {1243,2314}. By considering how theHasse graphs of permutations in these classes can be built from a sequence ofrooted source graphs, we determine their algebraic generating functions. Ourapproach is similar to that of "adding a slice", used previously to enumeratevarious classes of polyominoes and other combinatorial structures. To solve therelevant functional equations, we make extensive use of the kernel method.

The permutation classes Av(1234,2341) and Av(1243,2314)