Séminaire Lotharingien de Combinatoire, B59c (2008), 17 pp.
Number of "udu"s of a Dyck Path and ad-Nilpotent Ideals of
Parabolic Subalgebras of sll+1(C)
For an ad-nilpotent ideal i of a Borel subalgebra of
sll+1(C), we denote by
Ii the maximal subset I of the set of simple roots such that i is an ad-nilpotent ideal of the
standard parabolic subalgebra pI.
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
sll+1(C) and the set of Dyck paths of length
2l+2, to exhibit a bijection between ad-nilpotent ideals i of the
Borel subalgebra such that #Ii=r and the Dyck paths of
length 2l+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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