Title: Order in disordered packings with and without permutation symmetry

Abstract: A disordered solid, such as an athermal jammed packing of soft spheres, exists in a rugged potential-energy landscape in which there are a myriad of stable configurations that defy easy enumeration and characterization. Nevertheless, in three-dimensional monodisperse particle packings, we demonstrate an astonishing regularity in the distribution of basin volumes. The probability of landing randomly in a basin is proportional to its volume. Ordering the basins according to their probability, $P(n)$, from the largest at $n=1$ to smaller at larger $n$, we find approximately: $P(n) \propto n^{-1}$. This order, persisting up to the largest systems for which we can collect sufficient data, has implications for the dynamics of a system as it evolves under perturbations. In monodisperse packings there is ``permutation symmetry'' since identical particles can always be interchanged without affecting the system or its properties. Introducing any distribution of radii breaks this symmetry and leads to a proliferation of distinct configurations. We present an algorithm that partially restores permutation symmetry to such polydisperse packings.