Sources page biographical material

5.L. MÉNAGE PROBLEM
How many ways can n couples be seated, alternating sexes, with no couples adjacent?
A. Cayley. On a problem of arrangements. Proc. Roy. Soc. Edin. 9 (1878) 338 342. Problem raised by Tait. Uses inclusion/exclusion to get a closed sum.

T. Muir. On Professor Tait's problem of arrangements. Ibid., 382 387. Uses determinants to get a simple n term recurrence.

A. Cayley. Note on Mr. Muir's solution of a problem of arrangement. Ibid., 388 391. Uses generating function to simplify to a usable form.

T. Muir. Additional note on a problem of arrangement. Ibid., 11 (1882) 187 190. Obtains Laisant's 2nd order and 4th order recurrences.

É. Lucas. Théorie des Nombres. Gauthier Villars, Paris, 1891; reprinted by Blanchard, Paris, 1958. Section 123, example II, p. 215 & Note III, pp. 491 495. Lucas appears not to have known of the work of Cayley and Muir. He describes Laisant's results. The 2nd order, non homogeneous recurrence, on pp. 494 495, is attributed to Moreau.

C. Laisant. Sur deux problèmes de permutations. Bull. Soc. Math. de France 19 (1890 91) 105 108. General approach to problems of restricted occupancy. His work yields a 2nd order non-homogeneous recurrence and homogeneous 3rd and 4th order recurrences. He cites Lucas, but says Moreau's work is unpublished.

H. M. Taylor. A problem on arrangements. Messenger of Math. 32 (1903) 60 63. Gets almost to Muir & Laisant's 4th order recurrence.

J. Touchard. Sur un problème de permutations. C. R. Acad. Sci. Paris 198 (1934) 631 633. Solution in terms of a complicated integral. States the explicit summation.

I. Kaplansky. Solution of the "problème des ménages". Bull. Amer. Math. Soc. 49 (1943) 784 785. Obtains the now usual explicit summation.

I. Kaplansky & J. Riordan. The problème de ménages. SM 12 (1946) 113 124. Gives the history and a uniform approach.

J. Touchard. Permutations discordant with two given permutations. SM 19 (1953) 109 119. Says he prepared a 65pp MS developing the results announced in 1934 and rediscovered in Kaplansky and in Kaplansky & Riordan. Proves Kaplansky's lemma on selections by finding the generating functions which involve Chebyshev polynomials. Obtains the explicit summation, as done by Kaplansky. Extends to more general problems.

M. Wyman & L. Moser. On the 'problème des ménages'. Canadian J. Math. 10 (1958) 468 480. Analytic study. Updates the history -- 26 references. Gives table of values for n = 0 (1) 65.

Jacques Dutka. On the 'Problème des ménages'. Math. Intell. 8:3 (1986) 18 25 & 33. Thorough survey & history -- 25 references.

Kenneth P. Bogart & Peter G. Doyle. Non sexist solution of the ménage problem. AMM 93 (1986) 514 518. 14 references.