A350558 -id:A350558 - cp's OEIS frontend

A343700 a(n) is the number of preference profiles in the stable marriage problem with n men and n women such that there are no pairs of people who rank each other first.

0, 2, 9984, 28419102720, 175302739963548794880, 5801674463718565478400000000000000, 2113937863028052653298578438638220083200000000000000, 15500609395854457241550377325238753195602871153217230602240000000000000000

Offset: 1

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

nonn

Two people who rank each other first are called soulmates. Thus, this sequence enumerates the profiles without soulmates.
This sequence is in contrast to the sequence A343698 which enumerates profiles with n pairs of soulmates.
The preference profiles with the most stable matchings do not have soulmates.

			For n=2, suppose A and B are the men and C and D are the women, then the only two possibilities are the following: a) A prefers C, C prefers B, B prefers D, and D prefers A; 2) A prefers D, D prefers B, B prefers C, and C prefers A.

Michael De Vlieger, Table of n, a(n) for n = 1..22
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.

Cf. A185141, A343698, A343699, A350558, A284458.

Table[Total[
  Table[(-1)^i Binomial[n, i]^2 (n - 1)!^(2 i) i! n!^(2 n - 2 i), {i,
    0, n}]], {n, 10}]

a(n) = sum(i=0, n, ((-1)^i * binomial(n, i)^2 * (n - 1)!^(2*i) * i! * n!^(2*n - 2*i))); \\ Michel Marcus, Jan 20 2023

a(n) = Sum_{i = 0..n} ((-1)^i * binomial(n, i)^2 * (n - 1)!^(2i) * i! * n!^(2n - 2i)).

a(n) = A350558(n)*A284458(n). - Dan Eilers, Jan 17 2023

A284458 Number of pairs (f,g) of endofunctions on [n] such that the composite function gf has no fixed point.

Original entry on oeis.org

1, 0, 2, 156, 16920, 2764880, 650696400, 210105425628, 89425255439744, 48588905856409920, 32845298636854828800, 27047610425293718239100, 26664178085975252011318272, 31009985808408471580603417296, 42017027730087624384021945067520

Offset: 0

Robert FERREOL, Mar 27 2017

nonn

Consider n boys and n girls, each boy secretly loving a girl and vice versa. The probability that no couple could be formed is a(n)/n^(2n).
a(n) is the number of pairs of binary matrices n X n (M, N), M having only one 1 per row, N having only one 1 per column, such that M + N has no coefficient equal to 2.
Limit_{n -> infinity} a(n)/n^(2n) = 1/e.

			For two boys A,B and two girls A',B', the a(2) possibilities are:
A loves A' who loves B who loves B' who loves A;
A loves B' who loves B who loves A' who loves A.

Robert Israel, Table of n, a(n) for n = 0..214
Math StackExchange, An unexpected application of non-trivial combinatorics, 2014.
Math StackExchange, Couple Probability, 2014.
Eric Weisstein's MathWorld, Confluent Hypergeometric Function of the Second Kind

Cf. A065440, A350558, A343700.

a:=n->add((-1)^k*binomial(n,k)^2*k!*n^(2*(n-k)),k=0..n):
seq(a(n),n=0..15);

Table[Sum[(-1)^k Binomial[n,k]^2 * k! * n^(2*(n - k)), {k, 0, n}], {n, 1, 15}] (* Indranil Ghosh, Mar 27 2017 *)
Table[HypergeometricU[-n, 1, n^2], {n, 1, 15}] (* Jean-François Alcover, Mar 29 2017 *)

a(n) = sum(k=0, n, (-1)^k * binomial(n,k)^2 * k! * n^(2*(n-k))); \\ Michel Marcus, Apr 04 2017

a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k)^2 * k! * n^(2*(n-k)).

a(n) = A343700(n)/A350558(n). - Dan Eilers, Jan 17 2023

cp's OEIS Frontend

A343700 a(n) is the number of preference profiles in the stable marriage problem with n men and n women such that there are no pairs of people who rank each other first.

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