A343698 a(n) is the number of preference profiles in the stable marriage problem with n men and n women such that there are n pairs of soulmates (people who rank each other first).

1, 2, 384, 40310784, 7608405715845120, 6419592322744320000000000000, 50709051409862934701619019776000000000000000, 6988904507653043786857875068352862005134308147200000000000000000

Offset

1,2

Comments

For such profiles, each person has exactly one valid partner: their soulmate. Consequently, there is only one stable matching: where the soulmates are married to each other.

For these profiles, the Gale-Shapley algorithm, whether it is men-proposing or women proposing, ends in one round.

This is a subsequence of A001013.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..23

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 - 1)!^(2n) * n!.

Example

For n = 3, there are 3! = 6 ways to pair the men and women into soulmate pairs, then 2! ways to finish each person's preference profile, making 6 * 2!^6 = 384 ways to set up the preference profiles.

Mathematica

Table[(n - 1)!^(2 n) n!, {n, 10}]

Crossrefs

Cf. A001013, A185141, A343699, A343700.

Sequence in context: A280281 A225096 A225111 * A176937 A092701 A321230

Adjacent sequences: A343695 A343696 A343697 * A343699 A343700 A343701

Keyword

nonn

Author

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

Status

approved