I find myself able to solve the text-book examples for the application of generating functions, however I am very much incapable of translating other problems into generating functions.
For example, let us attempt to count the number of permutations of numbers from 1 to n, such that no two numbers that are consecutive appear consecutive in the proper order.
Ex: 4 2 1 3 is okay for n=4, but 3 4 2 1 is not since 3 and 4 are consecutive and appear consecutively in that order.
This question has an answer on this site. I am not looking for this answer; I am trying to find guidelines to translate such a problem into analytic functions.
The difficulties I face, are due, I believe, to the fact that most standard problems that appear in generating function examples are simple variations of Cartesian products, where decisions of different elements can be made independently.
Using the Symbolic method did not work either, because I did not know how to synthesize a class description using the individual numbers as atoms.
If someone could answer this question, or at least provide insight into whether generating functions are suitable for the task at hand, I'd be very grateful.