

A351413


a(n) is the maximum number of stable matchings in the Latin Stable Marriage Problem of order n.


1




OFFSET

1,2


COMMENTS

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, nth powers of the cyclic group C2 and nth 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 pseudoLatin 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 RCequivalent Latin squares (A123234) instead of reduced Latin squares.


REFERENCES

C. Converse, Lower bounds for the maximum number of stable pairings for the general marriage problem based on the latin marriage problem, Ph. D. Thesis, Claremont Graduate School, Claremont, CA (1992) [Examples are from 6970].


LINKS

Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, Sequences of the Stable Matching Problem, arXiv:2201.00645 [math.HO], 2021 [Sections 3.7 and 4.2].


EXAMPLE

Maximal instance of order 2 with 2 stable matchings:
12
21
Maximal instance of order 3 with 3 stable matchings:
123
231
312
Maximal instance of order 4 with 10 stable matchings (group C2xC2):
1234
2143
3412
4321
Maximal instance of order 5 with 9 stable matchings:
12345
21453
34512
45231
53124
Maximal instance of order 6 with 48 stable matchings (Dihedral group):
123456
214365
365214
456123
541632
632541
Maximal instance of order 7 with 61 stable matchings:
1234567
2316745
3125476
4657312
5743621
6471253
7562134


CROSSREFS



KEYWORD

nonn,hard,more


AUTHOR



STATUS

approved



