  The simple permutations in two permutation classes --- the 321-avoiding
permutations and the skew-merged permutations --- are enumerated using a
uniform method. In both cases, these enumerations were known implicitly, by
working backwards from the enumeration of the class, but the simple
permutations had not been enumerated explicitly. In particular, the enumeration
of the simple skew-merged permutations leads to the first truly structural
enumeration of this class as a whole. The extension of this method to a wider
collection of classes namely grid classes of infinite paths is discussed.
