ad-nilpotent b-Ideals in sl(n) having a fixed class of nilpotence: Combinatorics and enumeration

We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of sl(n + 1, ℂ). We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence. We study the relationships between these results and the combinatorics of Dyck paths, based upon a remarkable bijection between ad-nilpotent ideals and Dyck paths. Finally, we propose a (q, t)-analogue of the Catalan number C_n. These (q, t)-Catalan numbers count, on the one hand, ad-nilpotent ideals with respect to dimension and class of nilpotence and, on the other hand, admit interpretations in terms of natural statistics on Dyck paths.