Geodesic
Number of "udu"s of a Dyck Path and ad-Nilpotent Ideals of Parabolic Subalgebras of sll+1(C)
Séminaire lotharingien de combinatoire, Tome 59 (2008-2010) 
Céline Righi. Number of "udu"s of a Dyck Path and ad-Nilpotent Ideals of Parabolic Subalgebras of sll+1(C). Séminaire lotharingien de combinatoire, Tome 59 (2008-2010). http://geodesic.mathdoc.fr/item/SLC_2008-2010_59_a1/
@article{SLC_2008-2010_59_a1,
     author = {C\'eline Righi},
     title = {Number of "udu"s of a {Dyck} {Path} and {ad-Nilpotent} {Ideals} of {Parabolic} {Subalgebras} of {sll+1(C)}},
     journal = {S\'eminaire lotharingien de combinatoire},
     year = {2008-2010},
     volume = {59},
     url = {http://geodesic.mathdoc.fr/item/SLC_2008-2010_59_a1/}
}
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Voir la notice de l'acte provenant de la source Séminaire Lotharingien de Combinatoire website

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.

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