Also called the ménage problem. In how many ways can 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 can be given in terms of Laisant's
recurrence formula
with
and
(Dörrie 1965, p. 33).
A closed form expression for in terms of a sum due to Touchard (1953) is
The first few values of for , 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 . The numbers can be considered a special case of a restricted rooks
problem.
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