A343699 a(n) is the number of preference profiles in the stable marriage problem with n men and n women with n - 1 pairs of soulmates (people who rank each other first).

0, 12, 9216, 2418647040, 913008685901414400, 1348114387776307200000000000000, 17038241273713946059743990644736000000000000000, 3522407871857134068576369034449842450587691306188800000000000000000

Offset: 1

Author

Tanya Khovanova and MIT PRIMES STEP Senior group, May 26 2021

Keywords

• nonn

Comments

Such profiles have exactly one stable matching, where soulmates are married to each other.

The men-proposing Gale-Shapley algorithm when used on these preference profiles will end in j rounds if the man in the non-soulmate pair ranks his partner as j-th. A similar statement is true for the women-proposing Gale-Shapley algorithm.

Examples

			When n = 2, there are 2 ways to pick the man in the soulmate pair and 2 ways to pick the woman in the soulmate pair. After this, the soulmate's preference profiles are fixed. There are 4 ways to complete the profiles for the other two people, but 1 of the ways creates a second pair of soulmates, which is forbidden. Thus, there are 12 profiles with exactly one pair of soulmates.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..23
• Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, Sequences of the Stable Matching Problem, arXiv:2201.00645 [math.HO], 2021.
• Wikipedia, Gale-Shapley algorithm.

Crossrefs

Cf. A185141, A343698, A343700.

Programs

Mathematica

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

Formula

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