Graphs of reduced words and some connections

Abstract

The family of graphs of reduced words of a certain sub-collection of permutations in the union \(\cup_{n\geq 4}\frak{S}_{n}\) of symmetric groups is investigated. The sub-collection 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. Some isomorphisms of these graphs with various combinatorial objects are established. Lastly, a link with the Poincar\’e polynomial of the integral cohomology ring of the Grassmannian \({\rm Gr}(2,n)\) is also given.