These are described in detail below; however, we note here that they are all, N (N -1)/2, the internuclear distances among N atoms, r ij that are transformed to other variable that we denote by y ij . We note that the "flow" to get to these variables and their symmetry properties starts with the 3N cartesian coordinates of all atoms. From these the internuclear distances are obtained. Then permutations are applied in the cartesians and the transformed interuclear distances are obtained. This is the fundamental mapping from permutations in cartesian coordinates to internuclear distances and from those to transformed variables. Then finally the PIPs are symmetric with respect to the transformed variables.