

A344671


a(n) is the total number of stable matchings for all possible preference profiles in the stable marriage problem with n men and n women such that there exists a married couple where the woman and the man rank each other last.


1




OFFSET

1,2


COMMENTS

A man and a woman who rank each other last and end up in a marriage are called a hellcouple. A stable matching cannot have more than one hellcouple.
Given a profile, if there exists a stable matching with a hellcouple, then all the stable matchings for this profile have the same hellcouple.
The GaleShapley algorithm (both menproposing and womenproposing) for such a profile needs at least n rounds to terminate.
A344670(n) is the number of preference profiles such that there exists a stable matching with a hellcouple.
This sequence is distinct from A344670 because in this sequence profiles are counted with their respective multiplicity if they yield multiple stable matchings with a hellcouple.


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.


EXAMPLE

For n = 2, each preference profile that has a hellcouple has exactly one stable matching, thus a(2) = A344670(2) = 4. For n > 2, this is no longer the case and a(n) > A344670(n).


CROSSREFS



KEYWORD

nonn,more


AUTHOR



STATUS

approved



