M\'enage Numbers and M\'enage Permutations

TL;DR

This paper explores the combinatorial properties of ménages permutations, establishing formulas linking ménages numbers to Catalan numbers and counting permutations by cycle number.

Contribution

It introduces new formulas connecting ménages numbers with Catalan numbers and provides cycle-based enumeration methods for ménages permutations.

Findings

Established formulas relating ménages numbers and Catalan numbers

Derived cycle-based counting formulas for ménages permutations

Enhanced understanding of the combinatorial structure of ménages permutations

Abstract

In this paper, we study the combinatorial structures of straight and ordinary m\'enage permutations. Based on these structures, we prove four formulas. The first two formulas define a relationship between the m\'enage numbers and the Catalan numbers. The other two formulas count the m\'enage permutations by number of cycles.