  We describe an inner product on the diagrams on which the Temperley-Lieb
algebra can be represented. We exhibit several constructions which are in
natural combinatorial bijection with these diagrams, which are generalizations
of various constructions counted by the Catalan numbers. We use a method
similar to the existing ones for orthogonalizing the Temperley-Lieb algebra to
construct an orthogonal basis for the vector space over these diagrams.
