  The number of distinguishable inherent structures of a liquid is the key
component to understanding the thermodynamics of glass formers. In the case of
hard potential systems such as hard discs, spheres and ellipsoids, an inherent
structure corresponds to a collectively jammed configuration. This work
develops a tiling based approach to counting inherent structures that
constructs packings by combining sets of elementary locally jammed structures
but eliminates those final packings that either, do not tile space, or are not
collectively jammed, through the use of tile incompatibility rules. The
resulting theory contains a number of geometric quantities, such as the number
of growth sites on a tile and the number of tile compatibilities that provide
insight into the number of inherent structures in certain limits. We also show
that these geometric quantities become quite simple in a system of highly
confined hard discs.
