This paper presents a combinatorial study of the power contamination problem, a dynamic variant of power domination modeled on grid graphs. We resolve a conjecture posed by Ainouche and Bouroubi (2021) by proving it is false and instead establish the exact value of the power contamination number on grid graphs. Furthermore, we derive recurrence relations for this number and initiate the enumeration of optimal contamination sets. We prove that the number of optimal solutions for specific grid families corresponds to well-known integer sequences, including those counting ternary words with forbidden subwords and the large Schröder numbers. This work settles the fundamental combinatorial questions of the power contamination problem on grids and reveals its rich connections to classical combinatorics.

翻译：本文对电力污染问题进行了组合学研究，该问题是基于网格图建模的电力支配问题的一个动态变体。我们解决了Ainouche与Bouroubi（2021）提出的一个猜想，通过证明其错误，转而建立了网格图上电力污染数的精确值。此外，我们推导出该数的递推关系，并开启了最优污染集的枚举工作。我们证明，特定网格族的最优解数量对应于著名的整数序列，包括含有禁用子词的三元词计数序列和大Schröder数。本工作解决了网格上电力污染问题的基本组合学问题，并揭示了其与经典组合学之间的丰富联系。