A344670 a(n) is the number of preference profiles in the stable marriage problem with n men and n women such that there exists a stable matching with one couple where both the man and the woman rank each other last.

1, 4, 4536, 5113774080

Offset

1,2

Comments

A man and a woman who rank each other last and end up in a marriage are called a hell-couple. A stable matching cannot have more than one hell-couple.

Given a profile, if there exists a stable matching with a hell-couple, then all the stable matchings for this profile have the same hell-couple.

The Gale-Shapley algorithm (both men-proposing and women-proposing) for such a profile needs at least n rounds to terminate.

Links

Table of n, a(n) for n=1..4.

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.

Example

For n = 2, there is a stable matching with a hell-couple if and only if the other two people rank each other first. Now, there are 2 ways to pair the men and women, and 2 ways to choose which couple has a man and woman ranking each other first, making a(2) = 2 * 2 = 4.

Crossrefs

Cf. A185141, A344671.

Sequence in context: A079682 A127235 A274972 * A344671 A203036 A306962

Adjacent sequences: A344667 A344668 A344669 * A344671 A344672 A344673

Keyword

nonn,more

Author

Tanya Khovanova and MIT PRIMES STEP Senior group, Jun 02 2021

Status

approved