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.

1, 4, 576, 26873856, 1585084524134400, 320979616137216000000000000, 493004666484778531821296025600000000000000, 11093499218496894899774404870401368262117949440000000000000000

Offset: 1

Author

Tanya Khovanova and MIT PRIMES STEP Senior group, Apr 16 2021

Keywords

• nonn

Comments

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.

Examples

			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.
		

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.

Crossrefs

Cf. A185141, A342573, A340890.

Programs

Mathematica

Table[n^2 (n - 1)!^(2n), {n, 10}]

Formula

a(n) = n^2 * (n-1)!^(2*n).

a(n) = A342573(n)^2, where A342573 ignores women's preferences.