AI 中文总结
研究数独网格的计数、对称性与等价类问题，提出用列划分的无序三元组同构类这一不变量代替经典约简链，通过手动应用伯恩赛德引理，无需计算枚举得到\(44\)及经典约简的中间阶段。
AI 中文摘要
数独是一种广受欢迎的谜题，其完整网格已通过计算枚举：大约有\(6.67×10^{21}\)个，在对称和数字重命名下，本质上不同的有\(5,472,730,538\)个。经典枚举通过一系列特殊约简将计数减少到第一宫的\(44\)个等价类，但数字\(44\)缺乏明显结构解释。我们给出了这\(44\)个类的另一种推导，它们作为列划分的无序三元组（多重集）在重新标记下的同构类出现，一个单一不变量取代了原来的约简链。此不变量使手动应用伯恩赛德引理成为可能：通过无需计算枚举的封闭推导，我们得到了\(44\)以及经典约简的两个中间阶段。
英文摘要
Sudoku is a widely popular puzzle whose complete grids have been enumerated computationally: there are approximately $6.67 \times 10^{21}$ of them and, up to symmetry and renaming of digits, $5,472,730,538$ essentially different ones. The classical enumeration reduces the count to $44$ equivalence classes of the first band through a chain of ad hoc reductions, leaving the number $44$ without any apparent structural explanation. We present an alternative derivation of these $44$ classes, in which they arise as isomorphism classes of unordered triples (multisets) of column partitions under relabeling, a single invariant that replaces the original chain of reductions. This invariant makes it possible to apply Burnside's Lemma by hand: we recover $44$ through a closed derivation requiring no computational enumeration.
Comments24 pages, 3 tables
