%I #7 Jan 27 2022 23:16:33
%S 1,4,72,13824,19353600,585252864000,309856276316160000,
%T 4385849628750224818176000,2004821822925413274697731145728000,
%U 36224269774123479086515914989912457216000000,31014029806101314488308034499720939299674343342080000000
%N a(n) is the number of joint preference profiles in a stable marriage problem with n men and n women.
%C A joint profile is defined as a preference profile with a fixed function f(k), such that for every man ranked k by a woman, he has to rank her back as f(k).
%C In such a profile the ranking matrix of women's preferences forms a Latin square. The same is true for men's preferences.
%F a(n) = n!*A002860(n)
%e For n=2, there are two types of joint profiles. In the first type, the mutual rankings are (1,1) and (2,2). In the second type, the mutual rankings are (1,2) and (2,1). For each type, the profile is uniquely defined after choosing the woman to be the first choice for the first man (there are two ways to do it). Therefore, a(2) = 4.
%Y Cf. A185141, A002860.
%K nonn
%O 1,2
%A _Tanya Khovanova_ and MIT PRIMES STEP Senior group, Jun 11 2021
