  We introduce bijections between generalized type $A_n$ noncrossing partitions
(that is, associated to arbitrary standard Coxeter elements) and fully
commutative elements of the same type. The latter index the diagram basis of
the classical Temperley-Lieb algebra, while for each choice of standard Coxeter
element the corresponding noncrossing partitions also index a basis, given by
the images in the Temperley-Lieb algebra of the simple elements of the dual
Garside structure (associated to this choice of standard Coxeter element) of
the Artin braid group on $n+1$ strands. We then show that our bijections come
from triangular base changes between the diagram basis and the various bases
indexed by noncrossing partitions, by explicitly describing the orders giving
triangularity. These orders were introduced in a joint paper with Williams and
provide exotic lattice structures on noncrossing partitions. Several
combinatorial objects are introduced along the way, including an involution on
the set of noncrossing partitions.
