We study the combinatorics of
a
d
ad
-nilpotent ideals of a Borel subalgebra of
s
l
(
n
+
1
,
C
)
sl(n+1,\mathbb 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
a
d
ad
-nilpotent ideals and Dyck paths. Finally, we propose a
(
q
,
t
)
(q,t)
-analogue of the Catalan number
C
n
C_n
. These
(
q
,
t
)
(q,t)
-Catalan numbers count, on the one hand,
a
d
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.