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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
1, 4, 4608, 5317484544
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.
A344670(n) is the number of preference profiles such that there exists a stable matching with a hell-couple.
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 hell-couple.
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 hell-couple 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
Sequence in context: A127235 A274972 A344670 * A203036 A306962 A102205
KEYWORD
nonn,more
AUTHOR
Tanya Khovanova and MIT PRIMES STEP Senior group, Jun 05 2021
STATUS
approved