Ménage problem
In combinatorial mathematics, the ménage problem or problème des ménages asks for the number of different ways in which it is possible to seat a set of male-female couples at a dining table so that men and women alternate and nobody sits next to his or her partner. This problem was formulated in 1891 by Édouard Lucas and independently, a few years earlier, by Peter Guthrie Tait in connection with knot theory. For a number of couples equal to 3, 4, 5, ... the number of seating arrangements is
Mathematicians have developed formulas and recurrence equations for computing these numbers and related sequences of numbers. Along with their applications to etiquette and knot theory, these numbers also have a graph theoretic interpretation: they count the numbers of matchings and Hamiltonian cycles in certain families of graphs.
Let Mn denote the number of seating arrangements for n couples. Touchard (1934) derived the formula
Much subsequent work has gone into alternative proofs for this formula and into generalized versions of the problem that count seating arrangements in which some couples are permitted to sit next to each other. A different formula for Mn involving Chebyshev polynomials was given by Wyman & Moser (1958).
Until the work of Bogart & Doyle (1986), solutions to the ménage problem took the form of first finding all seating arrangements for the women and then counting, for each of these partial seating arrangements, the number of ways of completing it by seating the men away from their partners. However, as Bogart and Doyle showed, Touchard's formula may be derived directly by considering all seating arrangements at once rather than by factoring out the participation of the women.
There are 2×n! ways of seating the women: there are two sets of seats that can be arranged for the women, and there are n! ways of seating them at a particular set of seats. For each seating arrangement for the women, there are
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