Abstract
The separable permutations are those that can be obtained from the trivial permutation by two operations called direct sum and skew sum. This class of permutations contains the class of stack sortable permutations, Av(231), which are enumerated by the Catalan numbers. We prove that all subclasses of the separable permutations which do not contain Av(231) or a symmetry of this class have rational generating functions. Our principal tools include partial well-order (the lack of an infinite antichain), atomicity (the joint embedding property), and the theory of strongly rational permutation classes which is introduced here for the first time.