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 sll+1(C)

Abstract. 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. The following versions are available: