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.

a(n) is from Appendix C of Thurber's 2002 paper, using the maximum from each row. At the time of publication, the bounds were known to be exact up to n=4. A357269 shows that they are also exact for n=5. This sequence is not to be confused with A069156, also from Thurber's Appendix C, which uses only the first column, making for looser bounds for n > 11. a(6), a(8), a(10), a(12), and a(16) are also conjectured to be exact.

Improved lower bounds for n=7, 9, 11, 13, 15 are shown in linked Ong et al. (2025) file.

a(n) is the number of dihedral groups of order 2n with a fixed identity, or equivalently the number of reduced Latin squares of order 2n that can be viewed as the Cayley table of D_{2n}, by adding a border that matches the first row and column. The reduced Latin squares differ from each other by a permutation of their symbols. Two such Latin squares that differ by a permutation of their symbols have been called isoplanar by Bailey (1984), cited by Nilrat and Praeger (1988), cited by Denes and Keedwell (1991). Latin squares based on dihedral groups are of interest in the stable marriage problem, where Benjamin et al. (1995) exhibited such squares having many stable matchings when viewed as ranking matrices. Two isoplanar Latin squares generally produce a different number of stable matchings, so there is motivation to generate all symbol permutations to find the ones with the most stable matchings.

See comments in A002618 regarding automorphisms of dihedral groups by Ola Veshta and Yaghoub Sharifi. - Dan Eilers, Jun 08 2024

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.