I'm still wondering if there are simple correspondences between factorial representations and permutation polynomials. For simplicity, let's focus only on a prime # of elements. For example, over GF(p) there are p! permutations. Is there a polynomial form in which we have exactly p choices for one of the coefficients, exactly p-1 choices for the next coefficient, exactly p-2 choices for the next coefficient, and so on? What if the p basis functions are things other than monomials x, x^2, x^3, etc. ? With function composition, we can do the following: f(g(h(...(q(x))...))) can be a representation of one of these p! permutations if f(x) chooses one of p elements; g(x) chooses one of p-1 elements; h(x) chooses one of p-2 elements, and so forth. A form such as this should be relatively easy to manufacture. So it should be possible to represent any permutation in this fully-nested "standard" form. Are there forms involving multiplication instead of composition? Are there forms involving addition instead of composition?