Meanders and the Temperley-Lieb algebra

TL;DR

This paper explores the connection between meander statistics and the Temperley-Lieb algebra, deriving formulas for meander numbers through algebraic and combinatorial methods involving Gram determinants and random walks.

Contribution

It introduces a novel algebraic framework linking meander enumeration to the Temperley-Lieb algebra and provides explicit formulas for meander numbers using Gram determinants.

Findings

Derived the Gram determinant as a function of weight q.

Expressed meander numbers as sums over correlated random walks.

Established a new algebraic approach to meander enumeration.

Abstract

The statistics of meanders is studied in connection with the Temperley-Lieb algebra. Each (multi-component) meander corresponds to a pair of reduced elements of the algebra. The assignment of a weight $q$ per connected component of meander translates into a bilinear form on the algebra, with a Gram matrix encoding the fine structure of meander numbers. Here, we calculate the associated Gram determinant as a function of $q$, and make use of the orthogonalization process to derive alternative expressions for meander numbers as sums over correlated random walks.