  We introduce a new basis of the Temperley-Lieb algebra. It is defined using a
bijection between noncrossing partitions and fully commutative elements
together with a basis introduced by Zinno, which is obtained by mapping the
simple elements of the Birman-Ko-Lee braid monoid to the Temperley-Lieb
algebra. The combinatorics of the new basis involve the Bruhat order restricted
to noncrossing partitions. As an application we can derive properties of the
coefficients of the base change matrix between Zinno's basis and the well-known
diagram or Kazhdan-Lusztig basis of the Temperley-Lieb algebra. In particular,
we give closed formulas for some of the coefficients of the expansion of an
element of the diagram basis in the Zinno basis.
