cp's OEIS Frontend

Finding a(n) (denoted f(n) in the literature) is a research problem posed by Knuth in 1976 and reiterated by Gusfield and Irving in 1989.

a(5)=16 was found by Eilers using a MiniZinc constraint satisfaction model, showing previous lower bound of Eilers reported by Thurber to be exact.

A357271 gives lower bounds for a(n) reported by Thurber, previously known to be exact up to n=4, which is not to be confused with A069156 which gives looser lower bounds for n > 11.

Thurber proved a(n) to be strictly increasing.

There are 176130 reduced instances of order 5 with 16 stable matchings, and 498320 reduced instances with 15 stable matchings, compared with A351430 for order 4, and A369597 for order 3.

The total number of reduced instances for order n is A351409(n), which is 214990848000000000 for order 5, so about one in 1.22*10^12 such instances are maximal.

The maximum number of stable matchings for order 5, where the men's (or women's, respectively) ranking table is a latin square, is 14, with 300 such reduced instances, making order 5 the first order not containing any maximal instances where the men's ranking table is a Latin square.

"Reduced" instances are counted, as in A351430 for order 4.

Reduced instances are fewer than all instances by a factor of n!(n-1)! due to participant-renaming isomorphism, analogous to reduced latin squares.

The counts are listed in reverse order from A351430 so that the known terms appear first. The first term gives the number of maximal instances (with 16 stable matchings, per A357269).

The total number of reduced instances for order 5 is 214990848000000000, given by A351409(5).

The corresponding sequence for order 3 is [91,957,2840], with 91 maximal instances.

a(0) = A368433(5) = A344669(5) / (5! * 4!).

Results were obtained using the MiniZinc constraint modeling language, by extending the "Model for the Stable Marriage Problem" given in the MiniZinc Handbook, Section 2.2.6.

The first two terms were obtained on a PC with 12GB RAM; the next three terms on a PC with 64GB RAM.

Reduced instances (A351409) are fewer than all instances by a factor of n!(n-1)! due to participant-renaming isomorphism, analogous to reduced latin squares.

For n in [1,2,4], a(n) = 1 showing uniqueness up to isomorphism.

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.