%I #8 Feb 11 2022 11:52:35

%S 1,2,384,40310784,7608405715845120,6419592322744320000000000000,

%T 50709051409862934701619019776000000000000000,

%U 6988904507653043786857875068352862005134308147200000000000000000

%N a(n) is the number of preference profiles in the stable marriage problem with n men and n women such that there are n pairs of soulmates (people who rank each other first).

%C For such profiles, each person has exactly one valid partner: their soulmate. Consequently, there is only one stable matching: where the soulmates are married to each other.

%C For these profiles, the Gale-Shapley algorithm, whether it is men-proposing or women proposing, ends in one round.

%C This is a subsequence of A001013.

%H Michael De Vlieger, <a href="/A343698/b343698.txt">Table of n, a(n) for n = 1..23</a>

%H Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, <a href="https://arxiv.org/abs/2201.00645">Sequences of the Stable Matching Problem</a>, arXiv:2201.00645 [math.HO], 2021.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gale%E2%80%93Shapley_algorithm">Gale-Shapley algorithm</a>.

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

%e For n = 3, there are 3! = 6 ways to pair the men and women into soulmate pairs, then 2! ways to finish each person's preference profile, making 6 * 2!^6 = 384 ways to set up the preference profiles.

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

%Y Cf. A001013, A185141, A343699, A343700.

%K nonn

%O 1,2

%A _Tanya Khovanova_ and MIT PRIMES STEP Senior group, May 26 2021