Ménage problem

Posted on by Brandon Klein

In combinatorial mathematics, the ménage problem or problème des ménages[1] 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.[2] For a number of couples equal to 3, 4, 5, ... the number of seating arrangements is

    12, 96, 3120, 115200, 5836320, 382072320, 31488549120, ... (sequence A059375 in OEIS).

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.