The probl\`eme des m\'enages revisited

TL;DR

This paper offers a simpler, more elegant proof of the classic ménage problem formula, addressing gender bias issues in previous proofs and contributing to combinatorial mathematics.

Contribution

It provides an alternative, streamlined proof of the Touchard-Kaplansky formula that avoids gender bias present in earlier proofs.

Findings

New proof simplifies understanding of the problem

Addresses gender bias in combinatorial proofs

Enhances mathematical elegance and clarity

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).