A343697 a(n) is the number of preference profiles in the stable marriage problem with n men and n women such that both the men's and women's profiles form Latin squares.

1, 4, 144, 331776, 26011238400, 660727073341440000, 3779719071732351369216000000, 11832225237539469009819996424230666240000, 30522879094287825948996777484664523152536511038095360000, 99649061600109839440372937690884668992908741561885362729330828902400000000

Offset

1,2

Comments

Equivalently, these are the profiles where each woman is ranked differently by the n men and each man is ranked differently by the women.

The men-proposing Gale-Shapley algorithm on such a set of preferences ends in one round, since every woman receives one proposal in the first round. Similarly, the women-proposing Gale-Shapley algorithm ends in one round.

Links

Table of n, a(n) for n=1..10.

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.

Wikipedia, Gale-Shapley algorithm.

Formula

a(n) = A002860(n)^2.

Example

There are 12 Latin squares of order 3, where 12 = A002860(3). Thus, for n = 3, there are A002860(3) ways to set up the men's profiles and A002860(3) ways to set up the women's profiles, making A002860(3)^2 = 144 ways to set up all the preference profiles.

Crossrefs

Cf. A002860, A185141, A343696.

Sequence in context: A203424 A055209 A239350 * A030450 A041629 A278845

Adjacent sequences: A343694 A343695 A343696 * A343698 A343699 A343700

Keyword

nonn

Author

Tanya Khovanova and MIT PRIMES STEP Senior group, May 26 2021

Status

approved