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Séminaire Lotharingien de Combinatoire, B59c (2008), 17 pp.

# Céline Righi

# Number of "*udu*"s of a Dyck Path and *ad*-Nilpotent Ideals of
Parabolic Subalgebras of *sl*_{l+1}(**C**)

**Abstract.**
For an ad-nilpotent ideal *i* of a Borel subalgebra of
*sl*_{l+1}(**C**), we denote by
*I*_{i} the maximal subset *I* of the set of simple roots such that *i* is an ad-nilpotent ideal of the
standard parabolic subalgebra *p*_{I}.
We use the bijection of
Andrews, Krattenthaler, Orsina and Papi
[*Trans. Amer. Math. Soc.* **354** (2002), 3835-3853]
between the set of ad-nilpotent ideals of a Borel subalgebra in
*sl*_{l+1}(**C**) and the set of Dyck paths of length
2*l*+2, to exhibit a bijection between ad-nilpotent ideals *i* of the
Borel subalgebra such that #*I*_{i}=*r* and the Dyck paths of
length 2*l*+2 having *r* occurrences of "*udu*". We obtain
also a duality between antichains of cardinality *p* and *l*-*p*
in the set of positive roots.

Received: January 29, 2008.
Accepted: April 8, 2008.
Final Version: April 15, 2008.
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