Catalan numbers: from FC elements to classical diagram algebras

Sadek Al Harbat

Published 2023-08-19Version 1

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 definition of the Catalan numbers $C_{n+1}$ and find the Narayana numbers as well as the Catalan triangle. We determine the unique bijection between $W^c(A_n)$ and the set of non crossing diagrams that respects the monomial diagrammatical multiplication in the $A_n$-type Temperley-Lieb algebra.