The m\'enage problem with a known mathematician

TL;DR

This paper provides a combinatorial solution to a variant of the m'enage problem involving a known mathematician, detailing seating arrangements with specific constraints and a fixed starting position.

Contribution

It introduces a novel approach to the m'enage problem by considering a known mathematician's fixed position and specific seating choices.

Findings

Derived explicit formulas for seating arrangements with a known mathematician.

Analyzed the impact of the mathematician's choice on the total number of arrangements.

Extended classical m'enage problem to incorporate fixed-position constraints.

Abstract

We give a solution of the following combinatorial problem: "Let one from $n$ married couples in the m\'enage problem (see Problem 1) be a couple of a known mathematician $M$ and his wife. After the ladies are seated at every other chair, $M$ (in token of respect) is the first man allowed to choose one of the remaining chairs. To find the number of ways of seating the other men, with no man seated next to his wife, if $M$ chooses the chair that is $d$ seats clockwise from his wife's chair."