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.

翻译：在各种应用领域，如流体、细胞或人群模拟中，空间数据结构至关重要。它们用于回答最近邻查询，这些查询是执行必要计算（例如，模拟中推进至下一个时间步）的关键工具。相应地，研究人员开发了多种此类数据结构，其中一种为\emph{邻域网格}。本文研究该数据结构的组合学性质。特别地，我们证明先前工作中关于唯一性的假设实际上并不成立。我们将邻域网格的概念推广至任意网格尺寸与维度，并为因该不成立假设而失效的证明提供两种正确的替代版本。此外，我们探讨了该数据结构中特定状态的唯一性，以及这些状态数量何时达到最大值。通过使用矩形杨表的钩长公式，我们给出了局部分类。最后，我们提出了将此结果推广至所有二维情况的猜想。