Regular anti-phase templates in the stable marriage problem: a generator criterion, its converse, and a counting bound

Abstract

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.