  In various application fields, such as fluid-, cell-, or crowd-simulations,
spatial data structures are very important. They answer nearest neighbor
queries which are instrumental in performing necessary computations for, e.g.,
taking the next time step in the simulation. Correspondingly, various such data
structures have been developed, one being the \emph{neighborhood grid}.
  In this paper, we consider combinatorial aspects of this data structure.
Particularly, we show that an assumption on uniqueness, made in previous works,
is not actually satisfied. We extend the notions of the neighborhood grid to
arbitrary grid sizes and dimensions and provide two alternative, correct
versions of the proof that was broken by the dissatisfied assumption.
  Furthermore, we explore both the uniqueness of certain states of the data
structure as well as when the number of these states is maximized. We provide a
partial classification by using the hook-length formula for rectangular Young
tableaux. Finally, we conjecture how to extend this to all 2-dimensional cases.
