Graphs of Reduced Words and Some Connections

TL;DR

This paper studies a family of permutation graphs derived from reduced words with specific cycle structures, providing formulas for their vertices, degrees, and connections to other combinatorial objects.

Contribution

It introduces a new class of graphs based on permutations with hook cycle types, offering explicit formulas and isomorphisms to known combinatorial structures.

Findings

Closed formula for counting vertices of the graphs

Vertex-degree polynomials and generating series derived

Identified isomorphisms with various combinatorial objects

Abstract

The family of graphs of reduced words of a certain subcollection of permutations in the union $\cup_{n\geq 4}\frak{S}_{n}$ of symmetic groups is investigated. The subcollection is characterised by the hook cycle type $(n-2,1,1)$ with consecutive fixed points. A closed formula for counting the vertices of each member of the family is given and the vertex-degree polynomials for the graphs with their generating series is realised. Lastly, some isomorphisms of these graphs with various combinatorial objects are established.