Graphs of Reduced Words and Some Connections
Praise Adeyemo

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 of symmetic groups is investigated. The subcollection is characterised by the hook cycle type 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.
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Taxonomy
TopicsAdvanced Graph Theory Research
