Pattern avoidance classes and subpermutations
  Pattern avoidance classes of permutations that cannot be expressed as unions of proper subclasses can be described as the set of subpermutations of a single bijection. In the case that this bijection is a permutation of the natural numbers a structure theorem is given. The structure theorem shows that the class is almost closed under direct sums or has a rational generating function. 
M.D. Atkinson, M.M. Murphy, N. Ruskuc