稳定婚姻问题中的正则反相模板：生成元准则、其逆命题及计数界
Regular anti-phase templates in the stable marriage problem: a generator criterion, its converse, and a counting bound
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中文总结 AI 辅助
研究由有限群正则作用构建的稳定婚姻问题的高度对称实例，定义正则反相模板P(G,A)，证明生成元准则及其逆命题，建立稳定匹配数计数下界，还表明循环配置结构取决于排序，计算断言由脚本验证。
中文摘要 AI 辅助
我们研究了由有限群的正则作用构建的稳定婚姻问题的一类高度对称实例。给定阶为n的有限群G及其元素的一个排序A，我们定义正则反相模板P(G,A)。这些模板有n个典范稳定匹配。我们表明反相条件是典范的：在自同构极大配置中，反相模板恰好是那些满足常数秩和恒等式的模板。这给出了一个结构特征而非临时定义。我们证明了一个生成元准则及其精确逆命题：稳定集大小为n当且仅当每个相邻商群生成该群。此结果对所有有限群成立且无需交换性。我们进一步根据子群指标建立了稳定匹配数的计数下界。该界对于阶至多为5的群以及所有阶为4的群是紧的；特别地，对于克莱因群它给出至少10个稳定匹配，通过枚举确认等式成立。最后，我们表明循环配置并不总是产生链；结构取决于排序。所有计算断言都由一个随附脚本验证。
英文摘要
We study a family of highly symmetric instances of the stable marriage problem built from regular actions of finite groups. Given a finite group G of order n and an ordering A of its elements, we define the regular anti-phase template P(G,A). These templates have n canonical stable matchings. We show that the anti-phase condition is canonical: among automorphism-maximal profiles, the anti-phase templates are exactly those satisfying a constant rank-sum identity. This gives a structural characterization rather than an ad hoc definition. We prove a generator criterion and its exact converse: the stable set has size n if and only if each adjacent quotient generates the group. This result holds for all finite groups and does not require commutativity. We further establish a counting lower bound for the number of stable matchings in terms of subgroup indices. The bound is sharp for groups of order at most 5 and for all groups of order 4; in particular, it yields at least 10 stable matchings for the Klein group, with equality confirmed by enumeration. Finally, we show that cyclic profiles do not always produce chains; the structure depends on the ordering. All computational claims are verified by an accompanying script.
