  We investigate nil-Temperley-Lieb algebras of type A. We give a general
description of the structure of monomials formed by the generators. We also
show that the dimensions of these algebras are the famous Catalan numbers by
providing a bijection between the monomials and Dyck paths. We show that the
distribution of these monomials by degree is the same as the distribution of
Dyck paths by the sum of the heights of the peaks minus the number of peaks.
