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.

1, 27, 20736, 777600000, 2176782336000000, 645362587921121280000000, 27285016590396539545426329600000000, 213106813311662727500673631554480635904000000000, 386661002072680852777222237092449665784217600000000000000000000

Offset

2,2

Comments

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.

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.

Links

Table of n, a(n) for n=2..10.

Wikipedia, Gale-Shapley algorithm.

Formula

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

a(n) = A338665(n)/n!^(n) = sqrt(A343693(n)).

Example

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.

Mathematica

Table[n!^n/2^n, {n, 2, 10}]

Crossrefs

Cf. A185141, A338665, A343474, A343693.

Sequence in context: A036534 A167731 A357506 * A201369 A036529 A185882

Adjacent sequences: A343689 A343690 A343691 * A343693 A343694 A343695

Keyword

nonn

Author

Tanya Khovanova and MIT PRIMES STEP Senior group, May 23 2021

Status

approved