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This is also the number of preference profiles for n men and n women where every woman prefers man number 1 to man number 2.

When implementing the men-proposing Gale-Shapley algorithm on such a preference profile, woman number 1 gets her first engagement in an earlier round than woman number 2.

In the Latin Stable Marriage Problem of order n, the sum of a man and woman's rankings of each other is n+1. This implies that the men's and women's ranking tables are Latin squares. As a subproblem of the Stable Marriage Problem, Latin instances provide lower bounds for the maximum number of stable matchings in the general problem, such as A005154 and A065982. For sizes 1 to 4, Latin instances provide exact bounds; they are conjectured to provide exact bounds for sizes a power of 2; they provide the best lower bounds known for sizes 6, 10, 12, and 24, of 48, 1000, 6472, and 126112960, respectively.

The next term, a(8), is conjectured to be 268, consistent with A005154. The minimum number of stable matchings for Latin instances of order n is n, and is realized for the cyclic group of order n. The average number of stable matchings is 7 for n=4 (cf. A351430 showing an average of about 1.5 for the general problem), and benefits from avoidance of mutual first choices and more generally the lack of overlap between the men's and women's preferred matchings. The Latin squares of A005154 and A065982 can be interpreted as multiplication tables of groups, n-th powers of the cyclic group C2 and n-th dihedral groups, respectively.

The sequence decreases from a(4)=10 to a(5)=9, in contrast to the corresponding sequence for the general problem, which Thurber showed to be strictly increasing. This has motivated the study of less restrictive subproblems, such as pseudo-Latin squares (A069124, A069156), Latin x Latin instances (A344664, A344665, A343697), instances where participants have different first choices (A343475, A343694, A343695), or instances with unspecified/tied/template rankings (A284458 with only first choices specified).

The sequence is empirically derived, originally based on reduced Latin squares (A000315). There are fewer instances to try using RC-equivalent Latin squares (A123234) instead of reduced Latin squares.

a(n) is the number of women's ranking tables in the stable marriage problem that can be paired with a men's ranking table having no two men with the same first choice, without forming any mutual first choices. It has two terms: (n-1)^n from A065440(n), and (n-1)!^n from A091868(n-1). Such men's ranking tables having no two men with the same first choice arise in A343694, A343475, and A344663.

a(n)*A123234 is a useful alternative to A343696 which combines a Latin men's ranking table with an arbitrary women's table, since it gives fewer instances to consider.