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a(n) is the number of preference profiles for n men and n women where every man prefers woman number 1 to woman number 2.
2

%I #15 May 27 2021 06:56:01

%S 4,5832,6879707136,19349176320000000000,

%T 303256405652583481344000000000000,

%U 53311087345695615264200592956011315200000000000000,190584865366582887488321066784947980317795794157526056960000000000000000

%N a(n) is the number of preference profiles for n men and n women where every man prefers woman number 1 to woman number 2.

%C This is also the number of preference profiles for n men and n women where every woman prefers man number 1 to man number 2.

%C When implementing the men-proposing Gale-Shapley algorithm on such a preference profile, woman number 1 gets her first engagement in an earlier round than woman number 2.

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

%F a(n) = n!^(2n) / 2^n.

%e When n = 2, we have exactly 1 way to arrange each man's profiles such that woman number 1 is ranked before woman number 2. Each woman's profile can be set in 2! = 2 ways, so the total number of preference profiles such that every man prefers woman number 1 to woman number 2 is 1^2 * 2^2 = 4.

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

%Y Cf. A001013, A185141, A342573, A340890, A343474, A343475.

%K nonn

%O 2,1

%A _Tanya Khovanova_ and MIT PRIMES STEP Senior group, Apr 22 2021