Catalan numbers: from FC elements to classical diagram algebras

TL;DR

This paper explores the combinatorial structure of fully commutative elements in Coxeter groups of type A, connecting them to Catalan and Narayana numbers, and establishing a bijection with non-crossing diagrams in Temperley-Lieb algebras.

Contribution

It introduces a new bijection between fully commutative elements and non-crossing diagrams, with algorithms for conversion, linking algebraic and combinatorial structures.

Findings

Recovers Catalan numbers from the enumeration of fully commutative elements.

Identifies Narayana numbers and Catalan triangle through set partitions.

Establishes a bijection respecting diagrammatic multiplication in Temperley-Lieb algebra.

Abstract

Let $W^c(A_n)$ be the set of fully commutative elements in the $A_n$-type Coxeter group. Using only the settings of their canonical form, we recount $W^c(A_n)$ by the recurrence that is taken as a definition of the Catalan number $C_{n+1}$ and we find the Narayana numbers as well as the Catalan triangle via suitable set partitions of $W^c(A_n)$. We determine the unique bijection between $W^c(A_n)$ and the set of non-crossing diagrams of $n+1$ strings that respects the diagrammatic multiplication by concatenation in the $A_n$-type Temperley-Lieb algebra, along with the two algorithms implementing this bijection and its inverse.