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Let $W^c(A_n)$ be the set of fully commutative elements in the $A_n$-typeCoxeter 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 Catalannumber $C_{n+1}$ and we find the Narayana numbers as well as the Catalantriangle via suitable set partitions of $W^c(A_n)$. We determine the uniquebijection 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 implementingthis bijection and its inverse.

Catalan numbers: from FC elements to classical diagram algebras