Topic
Ménage problem
About: Ménage problem is a research topic. Over the lifetime, 4 publications have been published within this topic receiving 2 citations. The topic is also known as: Menage problem.
Papers
17 Aug 2016
TL;DR: This work proposes a novel approach based on enumeration of walks in certain algebraically weighted de Bruijn graphs that leads to new expressions for the menage numbers and their exponential generating function and allows one to efficiently compute the number of seating arrangements in general cases.
Abstract: We address the problem of enumeration of seating arrangements of married couples around a circular table such that no spouses sit next to each other and no k consecutive persons are of the same gender. While the case of \(k=2\) corresponds to the classical probleme des menages with a well-studied solution, no closed-form expression for the number of seating arrangements is known when \(k\ge 3\).
1 citations
TL;DR: In this paper, a solution to the menage problem is given through the use of largest rook polynomials, which are the polynomial whose coefficients have greatest sums which turn out to be Fibonacci numbers.
Abstract: A solution to the menage problem is given through the use of largest rook polynomials. These are rook polynomials whose coefficients have greatest sums which turn out to be Fibonacci numbers, while the sums of the coefficients of the rook polynomial for the reduced menage problem are Lucas numbers.
TL;DR: In this article, the problem of enumeration of seating arrangements of married couples around a circular table such that no spouses sit next to each other and no k consecutive persons are of the same gender is addressed.
Abstract: We address the problem of enumeration of seating arrangements of married couples around a circular table such that no spouses sit next to each other and no k consecutive persons are of the same gender. While the case of k=2 corresponds to the classical probleme des menages with a well-studied solution, no closed-form expression for the number of seating arrangements is known when k>=3.
We propose a novel approach for this type of problems based on enumeration of walks in certain algebraically weighted de Bruijn graphs. Our approach leads to new expressions for the menage numbers and their exponential generating function and allows one to efficiently compute the number of seating arrangements in general cases, which we illustrate in detail for the ternary case of k=3.
TL;DR: In this article, a finite-order system of recurrence relations for the permanent of circulant matrices containing a band of k any-value diagonals on top of a uniform matrix is presented.
Abstract: We present a finite-order system of recurrence relations for the permanent of circulant matrices containing a band of k any-value diagonals on top of a uniform matrix (for k=1,2 and 3) and the method for deriving such recurrence relations, which is based on the permanents of the matrices with defects. The proposed system of linear recurrence equations with variable coefficients provides a powerful tool for the analysis of the circulant permanents, their fast, linear-time computing; and finding their asymptotics in a large-matrix-size limit. The latter problem is an open fundamental problem. Its solution would be tremendously important for a unified analysis of a wide range of the nature’s ♯P-hard problems, including problems in the physics of many-body systems, critical phenomena, quantum computing, quantum field theory, theory of chaos, fractals, theory of graphs, number theory, combinatorics, cryptography, etc.
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2016 | 1 |
| 2015 | 1 |
| 1998 | 1 |