Generating permutations with restricted containers

We investigate a generalization of stacks that we call C- machines. We show how this viewpoint rapidly leads to functional equations for the classes of permutations that C-machines generate, and how these systems of functional equations can be iterated and sometimes solved. General results about the rationality, algebraicity, and the existence of Wilfian formulas for some classes generated by C-machines are given. We also draw attention to some relatively small permutation classes which, although we can generate thousands of terms of their counting sequences, seem to not have D-finite generating functions.