It is quite well known that the generation function for permutations is represented as $$(1+x)(1+x+x^2)\dots(1+x+x^2+x^3...+x^{n−1})$$ (See, e.g., question The generating function for permutations indexed by number of inversions)
I am looking for any examples to see application of the g.f. in geometry and particulary any geometrical visualization of the generation function.
Any ideas, problems, examples of visualization?