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

1,2

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.

Links

Table of n, a(n) for n=1..8.

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.

Formula

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

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

Example

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.

Mathematica

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

Crossrefs

Cf. A185141, A342573, A340890.

Sequence in context: A134372 A177926 A241095 * A177963 A102201 A102204

Adjacent sequences: A343471 A343472 A343473 * A343475 A343476 A343477

Keyword

nonn

Author

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

Status

approved