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
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
KEYWORD
nonn,more
AUTHOR
Tanya Khovanova and MIT PRIMES STEP Senior group, Jun 02 2021
STATUS
approved