A343696 a(n) is the number of preference profiles in the stable marriage problem with n men and n women, such that the men's preference profiles form a Latin square.

1, 8, 2592, 191102976, 4013162496000000, 113241608573209804800000000, 5078594244241245901264634511360000000000, 759796697672599288560347581750936194390876487680000000000, 602809439070636186475532789128702956081602819845966698324215778508800000000000

Offset

1,2

Comments

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

Equivalently, for every rank i, there is exactly one woman who is ranked i by a given man.

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.

Due to symmetry, a(n) is the number of preference profiles in the stable marriage problem with n men and n women, such that the women’s preference profiles form a Latin square.

Links

Table of n, a(n) for n=1..9.

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) = n!^n * A002860(n).

Example

For n = 3, there are 3!^3 ways to set up the women's preference profiles and A002860(3) ways to set up the men's preference profiles, where A002860(3) = 12 (there are 12 different Latin squares of order 3). Thus a(3) = 3!^3 * A002860(3) = 216 * 12 = 2592.

Crossrefs

Cf. A002860, A185141, A343697.

Sequence in context: A268150 A325062 A247733 * A210126 A259440 A172957

Adjacent sequences: A343693 A343694 A343695 * A343697 A343698 A343699

Keyword

nonn

Author

Tanya Khovanova and MIT PRIMES STEP Senior group, May 25 2021

Status

approved