发表机构
Toyohashi University of Technology（丰桥技术科学大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明稳定婚姻平衡实例存在唯一最细素块划分，同时分解执行图、反拟阵和格，并给出精确普查，发现可分解比例渐近为n!/n^(2n)，且划分可在多项式时间内计算。
AI 中文摘要
具有严格完全偏好的稳定婚姻问题的每个平衡实例都有一个唯一的最细划分为素块，且该单一划分同时分解三种不同结构：可达执行有向图作为笛卡尔积、提案前缀反拟阵作为直和、稳定匹配格作为直积。反之则不成立，且从规模二起在每个规模上都不成立：两个显式族共享相同的布尔立方体执行，而其中一个具有单个稳定匹配的最大可分解性，另一个则是具有n的素实例。唯一性产生一个精确普查，即一个递归计数每个规模下素实例的递归，据此，在每侧四个代理的110,075,314,176个配置中，恰好88,478,208个是可分解的，且可分解比例渐近为n! / n^(2n)。这些块被刻画为互秩过滤的平方分量，因此该划分可在多项式时间内计算，且该分解是一种工具而不仅仅是一个事实。
英文摘要
Every balanced instance of the stable marriage problem with strict complete preferences has a unique finest partition into prime blocks, and that single partition simultaneously factors three different structures: the reachable execution digraph as a Cartesian product, the proposal-prefix antimatroid as a direct sum, and the stable-matching lattice as a direct product. The converse fails, and fails at every size from two on: two explicit families share the identical Boolean-cube execution while one is maximally decomposable with a single stable matching and the other is prime with n. Uniqueness yields an exact census, a recursion counting the prime instances at every size, under which exactly 88,478,208 of the 110,075,314,176 profiles with four agents on each side are decomposable and the decomposable fraction is asymptotically n! / n^(2n). The blocks are characterised as the square components of the mutual-rank filtration, so the partition is computable in polynomial time and the factorisation is a tool rather than only a fact.
Comments29 pages, 1 figure. Ancillary files: an independent verification script that reproduces every exhaustive and exact count in the paper, and a cross-check of the two implementations
