Hey! I'm a postdoc at the University of Melbourne.
Did you know that the polynomial function f(x) = x3 + 6x2 + x permutes the elements of Z/9 (the integers modulo nine)? Not all polynomials permute the elements of the underlying coefficient ring. The ones that do are called permutation polynomials, and for finite coefficient rings the corresponding permutations form a group! Check out this recent paper to learn more about these polynomials.