Number of "udu" 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 $s{l}_{l+1}(C)$, 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 given by G.E. Andrews, C. Krattenthaler, L. Orsina and P. Papi between the set of ad-nilpotent ideals of a Borel subalgebra in $s{l}_{l+1}(C)$ and the set of Dyck paths of length $2l+2$, to explicit a bijection between ad-nilpotent ideals $i$ of the Borel subalgebra such that the cardinality of $I_i$ is equal to $r$ and the Dyck paths of length $2l+2$ having $r$ occurence "udu". We obtain also a duality between antichains of cardinality $p$ and $l-p$ in the set of positive roots.