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%I A343692 #11 May 26 2021 02:25:39
%S A343692 1,27,20736,777600000,2176782336000000,645362587921121280000000,
%T A343692 27285016590396539545426329600000000,
%U A343692 213106813311662727500673631554480635904000000000,386661002072680852777222237092449665784217600000000000000000000
%N A343692 a(n) is the number of men's preference profiles in the stable marriage problem with n men and n women, where every man prefers woman number 1 to woman number 2.
%C A343692 When implementing the men-proposing Gale-Shapley algorithm on such a preference profile, woman number 1's first engagement comes in an earlier round than the engagement of woman number 2.
%C A343692 This is the same as the number of women's preference profiles in the stable marriage problem with n men and n women, where every woman prefers man number 1 to man number 2.
%H A343692 Wikipedia, <a href="https://en.wikipedia.org/wiki/Gale%E2%80%93Shapley_algorithm">Gale-Shapley algorithm</a>.
%F A343692 a(n) = n!^(n) / 2^n.
%F A343692 a(n) = A338665(n)/n!^(n) = sqrt(A343693(n)).
%e A343692 When n = 2, there is exactly 1 way for each man's profile to be completed such that woman number 1 is before woman number 2. Since we are only looking at men's profiles, this means there are 1^n = 1^2 = 1 preference profiles such that every man prefers woman number 1 to woman number 2.
%t A343692 Table[n!^n/2^n, {n, 2, 10}]
%Y A343692 Cf. A185141, A338665, A343474, A343693.
%K A343692 nonn
%O A343692 2,2
%A A343692 _Tanya Khovanova_ and MIT PRIMES STEP Senior group, May 23 2021