I spent a lot of time today trying to write down, very specifically, what it means for a function to be invariant with respect to permutation of its input arguments. It turns out that this is hard! Especially when the function is a vector function of vector inputs. This is all related to our nascent NeurIPS submission. This symmetry, by the way, is the symmetry enforced by graph neural networks. But it is also a symmetry of all of classical physics (if, say, the vectors are the properties of particles).