Married Couples Problem

Also called the ménage problem. In how many ways can n married couples be seated around a circular table in such a manner than there is always one man between two women and none of the men is next to his own wife? The solution (Ball and Coxeter 1987, p. 50) uses discordant permutations and was solved by Laisant, Moreau, and Taylor. The solution for n>1 can be given in terms of Laisant's recurrence formula


with A_2=0 and A_3=1 (Dörrie 1965, p. 33).

A closed form expression for n>1 in terms of a sum due to Touchard (1953) is

 A_n=sum_(k=0)^n(2n)/(2n-k)(2n-k; k)(n-k)!(-1)^k,

where (n; k) is a binomial coefficient (Comtet 1974, p. 185; Vardi 1991, p. 123).

The first few values of A_n for n=2, 3, ... obtained from the recurrence and sum above are 0, 1, 2, 13, 80, 579, ... (OEIS A000179), which are sometimes called ménage numbers. The desired solution is then 2n!A_n. The numbers A_n can be considered a special case of a restricted rooks problem.

See also

Laisant's Recurrence Formula, Rooks Problem

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Married Couples Problem

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Weisstein, Eric W. "Married Couples Problem." From MathWorld--A Wolfram Web Resource.

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