The probl\`eme des m\'enages revisited

Abstract

We present an alternative proof to the Touchard-Kaplansky formula for the probl\`eme des m\'enages, which, we believe, is simpler than the extant ones and is in the spirit of the elegant original proof by Kaplansky (1943). About the latter proof, Bogart and Doyle (1986) argued that despite its cleverness, suffered from opting to give precedence to one of the genders for the couples involved (Bogart and Doyle supplied an elegant proof that avoided such gender-dependent bias).