Tiling bijections between paths and Brauer diagrams

TL;DR

This paper establishes a natural bijection between overhang paths, a generalization of Dyck paths, and basis diagrams of the Brauer algebra, extending previous tiling bijections with the Temperley-Lieb algebra.

Contribution

It introduces a new tiling bijection connecting overhang paths with Brauer algebra basis diagrams, broadening the combinatorial understanding of algebraic structures.

Findings

Bijection between overhang paths and Brauer diagrams established

Extension of tiling bijections from Dyck paths to overhang paths

Provides combinatorial framework for Brauer algebra basis diagrams

Abstract

There is a natural bijection between Dyck paths and basis diagrams of the Temperley-Lieb algebra defined via tiling. Overhang paths are certain generalisations of Dyck paths allowing more general steps but restricted to a rectangle in the two-dimensional integer lattice. We show that there is a natural bijection, extending the above tiling construction, between overhang paths and basis diagrams of the Brauer algebra.