This material has been published in
Trans. Amer. Math. Soc.
354
(2002), 3835-3853,
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ad-Nilpotent b-ideals
in sl(n) having a fixed class of nilpotence: Combinatorics and
enumeration
(19 pages)
Abstract.
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
Cn. 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.
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