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A343474 a(n) is the number of preference profiles for n men and n women, where all men prefer the same woman and all women prefer the same man. 6
1, 4, 576, 26873856, 1585084524134400, 320979616137216000000000000, 493004666484778531821296025600000000000000, 11093499218496894899774404870401368262117949440000000000000000 (list; graph; refs; listen; history; text; internal format)
Every preference profile of this type has exactly one pair of people who rank each other first.
This is the same number of preference profiles as when all men rank the same woman at only the i-th place, and all women rank the same man at only the j-th place, where i and j can be anywhere from 1 to n.
The total number of possible profiles is A185141.
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.
a(n) = n^2 * (n-1)!^(2*n).
a(n) = A342573(n)^2, where A342573 ignores women's preferences.
When n=2, there are 4 ways to pick a man and woman who are preferred by all people of the opposite gender, and then 1 way to fill in each of the remaining slots in every person's preference profile. So, there are 4 different preference profiles.
Table[n^2 (n - 1)!^(2n), {n, 10}]
Sequence in context: A134372 A177926 A241095 * A177963 A102201 A102204
Tanya Khovanova and MIT PRIMES STEP Senior group, Apr 16 2021

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Last modified May 18 09:53 EDT 2024. Contains 372619 sequences. (Running on oeis4.)