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\begin{document}

\title[Number of ``$udu$''s of a Dyck path and ad-nilpotent ideals]{Number of ``$udu$''s of a Dyck path and $ad$-nilpotent ideals of parabolic subalgebras of $sl_{\ell+1}(\cset)$}
\author{C\'eline RIGHI}
\address{UMR 6086 CNRS,  D\'epartement de Math\'ematiques, Universit\'e de Poitiers, T\'el\'eport 2 - BP
  30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope
  Chasseneuil Cedex, France}
\email{celine.righi@math.univ-poitiers.fr}



\begin{abstract}
For an ad-nilpotent ideal $\ilie $ of a Borel subalgebra of $sl_{\ell+1}(\cset)$, we denote by 
$I_{\ilie}$ the maximal subset $I$ of the set of simple roots such that $\ilie$ is an ad-nilpotent ideal of the 
standard parabolic subalgebra $\plie_I$. We use the bijection of 
Andrews, Krattenthaler, Orsina and Papi 
[{\it Trans.\ Amer.\ Math.\ Soc.} {\bf 354} (2002), 3835--3853]
between the set of ad-nilpotent ideals of a Borel subalgebra in $sl_{\ell+1}(\cset)$ and the set of Dyck paths of length $2\ell+2$, to exhibit a bijection between ad-nilpotent ideals $\ilie$ of the 
Borel subalgebra such that $\sharp I_{\ilie}=r$ and the Dyck paths of length $2\ell+2$ having $r$ occurrences of ``$udu$''. We obtain 
also a duality between antichains of cardinality $p$ and $\ell-p$ in the set of positive roots.
\end{abstract}

%\ms{17B20}

\maketitle

%First page headline in LaTeX for S\'eminaire Lotharingien de Combinatoire
%--first part
\thispagestyle{myheadings}
\font\rms=cmr8 
\font\its=cmti8 
\font\bfs=cmbx8

\markright{\its S\'eminaire Lotharingien de
Combinatoire \bfs 59 \rms (2008), Article~B59c\hfill}
\def\thepage{}

\section{Introduction}
Let $M_{{\ell}+1}(\cset)$ be the set of $({\ell}+1)$-by-$({\ell}+1)$ matrices with coefficients in $\cset$, and $\glie$ be the simple Lie algebra $sl_{{\ell}+1}(\cset)$ consisting of elements of $M_{{\ell}+1}(\cset)$ whose trace is equal to zero. 
Let $\hlie$ be the maximal toral subalgebra of $\glie$ consisting of trace zero diagonal matrices. Let $(E_{i,j})$ be the canonical basis 
of $M_{{\ell}+1}(\cset)$ and $(E_{i,j}^*)$ be its dual basis. For $1\leqslant i \leqslant {\ell}+1$, set $\epsilon_i=E_{i,i}^*$. Then $\Delta=\{\epsilon_i -\epsilon_j; 1\leqslant i,j\leqslant {\ell}+1, i\not= j\}$ is the root system associated to $(\glie,\hlie)$, and  $\Delta^+=\{\epsilon_i -\epsilon_j; 1\leqslant i<j\leqslant {\ell}+1\}$ is a system of positive roots. Denote by $\alpha_i=\epsilon_i-\epsilon_{i+1}$, for $i=1,\dots, {\ell}$. Then $\Pi=\{\alpha_1,
\dots, \alpha_{\ell}\}$ is the corresponding set of simple roots. For each $\alpha\in\Delta$, let 
$ \mathfrak{g}_{ \alpha}=\{x\in\glie; [h,x]=\alpha(h)x \mbox{ for all } h\in\hlie\}$ be the root space of $\glie$ relative to $\alpha$.

For  $I \subset \Pi$, set $\Delta_I=\zset I \cap\Delta$.
We fix the corresponding standard parabolic subalgebra,   
$$
\plie_I=\hlie\oplus\left(\bigoplus\limits_{\alpha \in \Delta_I \cup \Delta^+}\glie_{\alpha}\right).
$$
Note that $\plie_{\emptyset}$ is a Borel subalgebra $\blie$ associated to the choice of $\Delta^+$.

An ideal $\ilie $ of $\plie_I$ is ad-nilpotent if and only if for all $x\in \ilie$, $ad_{\plie_I} x$ is nilpotent. Since any ideal of $\plie_I$ is $\hlie$-stable, we can deduce easily that an ideal is ad-nilpotent if and only if it is nilpotent. Moreover, we have $\ilie =\bigoplus\limits_{\alpha \in \Phi} \glie_{\alpha}$, for some subset $\Phi\subset \Delta^+ \setminus \Delta_I$. 

A Dyck path of length $2n$ can be defined as a word of $2n$ letters $u$ or $d$, having the same number of $u$ and $d$, and such that there is always more $u$'s than $d$'s to the left of a letter.

%On the other hand, A Dyck path of length $2n$ is a path which begins at the origin $(0,0)$, ends at $(2n,0)$ %and consists of diagonal lines of direction $(1,1)$ and $(1,-1)$, such that the path stays above the line %%$x=0$. We can encode each $(1,1)$ by the letter $u$ (for up), each $(1,-1)$ by the letter $d$ (for down), %thus obtaining the encoding of a Dyck path by a word, called a Dyck word.

Andrews, Krattenthaler, Orsina and Papi established in \cite{AKOP} a bijection between the set of ad-nilpotent ideals of the Borel 
subalgebra $\plie_{\emptyset}$ and the set of Dyck paths of length $2\ell+2$ which allows them to enumerate ad-nilpotent ideals of a 
fixed class of nilpotence. The purpose of this paper is to explain some applications of this correspondence for the ad-nilpotent ideals of parabolic subalgebras.

%First page headline in AmS-LaTeX for S\'eminaire Lotharingien de Combinatoire
%--restoring the headers and pagenumbering
\pagenumbering{arabic}
\addtocounter{page}{1}
\markboth{\SMALL C\'ELINE RIGHI}{\SMALL NUMBER OF ``$udu$"S OF A DYCK PATH
AND ad-NILPOTENT IDEALS}
%
%

More precisely, let $\ilie$ be an ad-nilpotent ideal of the Borel subalgebra $\plie_{\emptyset}$. Denote by 
$I_{\ilie}$ the maximal subset $I\subset \Pi$ such that $\ilie$ is an ad-nilpotent ideal of $\plie_I$. The main result we prove here is the following theorem.

\begin{thm*}
There is a bijection between the ad-nilpotent ideals $\ilie$ of $\blie$ such that $\sharp I_{\ilie}=r$ and the Dyck paths of length $2\ell+2$ having $r$ 
occurrences of ``$udu$''.

\end{thm*}


We can deduce a formula for the desired number of ideals since the number of Dyck paths having $r$ occurrences of ``$udu$'' have been calculated in \cite{Su}. 


This paper is organized as follows: we first recall the natural bijection between $\ell$-partitions and Dyck paths of length $2\ell+2$, as in \cite{P}. In Section~3, we recall the iterative construction of the bijection of \cite{AKOP}. Then, in Section~4, we explain how to calculate the number of occurrences of ``$udu$'' of a Dyck path obtained by the previous construction. In Section~5, we recall some facts of \cite{R} and \cite{CP} on ad-nilpotent ideals and we prove Theorem~1. Finally, in Section~6, we establish a duality between ad-nilpotent ideals of $\plie_{\emptyset}$. Such a duality has already been constructed by Panyushev in \cite{P}, however, it is not the same as the one we have here.

\bigskip
\noindent
{\bf Acknowledgment.} This work was realized while I was visiting the Istituto Guido Castelnuvo di 
Matematica (Roma). I would like to thank the European program Liegrits for offering me the possibility to go there and the institute for its hospitality.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Partitions and Dyck paths}\label{partition_Dyck_path}
 
In this section, we shall see how to generate a Dyck path from a partition. 

Recall that a partition is an $\ell$-tuple $\lambda=(\lambda_1, \lambda_2,\dots, \lambda_{\ell})\in\nset^{\ell}$ such that $\lambda_1 \geqslant \lambda_2 \geqslant \dots\geqslant \lambda_{\ell}$. A partition will be called an $\ell$-partition if $\lambda_i\leqslant i$ for $i=1,\dots, l$.  

Partitions are usually represented by their Ferrers diagrams. Let $T_{\ell}$ be the Ferrers diagram of the ${\ell}$-partition $({\ell},{\ell}-1,\dots,1)$. Then the Ferrers diagram $F$ of any ${\ell}$-partition $\lambda$ can be viewed as a subdiagram of $T_{\ell}$. For example, for ${\ell}=5$, the Ferrers diagram of 
$\lambda=(3,1,1,0,0)$ is the subdiagram of $T_{\ell}$, whose boxes are denoted by some $\star$:
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\htrait&\htrait&\htrait&\htrait&\htrait\cr
\sboite{$\star$}&\sboite{$\star$}&\sboite{$\star$}&\sboite{}&%
\sboite{}&\vtrait\cr
\htrait&\htrait&\htrait&\htrait&\htrait\cr
\sboite{$\star$}&\sboite{}&\sboite{}%
&\sboite{}&\vtrait\cr
\htrait&\htrait&\htrait&\htrait\cr
\sboite{$\star$}&\sboite{}&\sboite{}&\vtrait\cr
\htrait&\htrait&\htrait\cr
\sboite{}&\sboite{}&\vtrait\cr
\htrait&\htrait\cr
\sboite{}&\vtrait\cr
\htrait\cr
}
}
}^{\ell}
$$

Let $\lambda=(\lambda_1,\dots,\lambda_{\ell})$ be an ${\ell}$-partition and let $F$ be its Ferrers diagram. We draw a dotted horizontal line from the top of the line $x+y={\ell}+1$ to $F$ and a dotted vertical line from $F$ to the bottom of the line $x+y={\ell}+1$. For example, when $\lambda=(5,3,1,1,1,0,0)$, we have:
$$
\begin{Tiles}{1}
\Dbloc{\Ttop\Tleft}\Dbloc{\Ttop}\Dbloc{\Ttop}\Dbloc{\Ttop\Tbottom}\Dbloc{\Ttop\Tbottom\Tright}\Dbloc{\Ttopleftdot\Ttopdots}\Dbloc{\Ttopleftdot\Ttopdots}\Dbloc{\Ttopleftdot\Ttopdots\Ttoprightdot\Tantidiagonal}\Dskip
\Dbloc{\Tleft}\Dbloc{\Tbottom}\Dbloc{\Tright\Tbottom}\Dspace\Dspace\Dbloc{}\Dbloc{\Tantidiagonal}\Dbloc{\Ddoubletext{c}{x+y=\ell+1}}\Dskip
\Dbloc{\Tleft\Tright}\Dbloc{}\Dbloc{}\Dbloc{}
\Dbloc{}\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft\Tright}\Dspace\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft\Tright\Tbottom}\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Ttopleftdot\Tleftdots}\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Ttopleftdot\Tleftdots}\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Ttopleftdot\Tleftdots\Tbottomrightdot\Tantidiagonal}\Dskip
\end{Tiles}
$$
\caption{}\label{fig_classique}


If we rotate the figure clockwise by $45$ degrees, we can easily see that we obtain  a Dyck path of length $2{\ell}+2$ called $P(\lambda)$ as in \cite{P}. This construction defines clearly a bijection $P: \lambda \mapsto P(\lambda)$ between ${\ell}$-partitions and Dyck paths of length $2{\ell}+2$. In the above example, the Dyck path $P(\lambda)$ is:
\begin{center}
$$
\Gitter(18,6)(0,0)
\Koordinatenachsen(18
,5)(0,0)
\Pfad(0,0),3334333443443444\endPfad
\Label\lu{0}(0,0)
\Label\u{\scriptstyle 1}(1,0)
\Label\u{\scriptstyle 2}(2,0)
\Label\u{\scriptstyle 3}(3,0)
\Label\u{\scriptstyle 4}(4,0)
\Label\u{\scriptstyle 5}(5,0)
\Label\u{\scriptstyle 6}(6,0)
\Label\u{\scriptstyle 7}(7,0)
\Label\u{\scriptstyle 8}(8,0)
\Label\u{\scriptstyle 9}(9,0)
\Label\u{\scriptstyle 10}(10,0)
\Label\u{\scriptstyle 11}(11,0)
\Label\u{\scriptstyle 12}(12,0)
\Label\u{\scriptstyle 13}(13,0)
\Label\u{\scriptstyle 14}(14,0)
\Label\u{\scriptstyle 15}(15,0)
\Label\u{\scriptstyle 16}(16,0)
\Label\l{\scriptstyle 1}(0,1)
\Label\l{\scriptstyle 2}(0,2)
\Label\l{\scriptstyle 3}(0,3)
\hskip10.5cm
$$
%\caption{$P(\lambda)$}\label{p(lambda)}
\end{center}

%----------------------------------------------------------------

\section{AKOP-bijection}\label{section_AKOP_bijection}


Let $\lambda=(\lambda_1,\dots,\lambda_{\ell})$ be an ${\ell}$-partition whose Ferrers diagram is $F$. We shall draw a dotted line associated to $\lambda$. We start at the top of the line $x+y={\ell}+1$. We go left until we meet $F$. Then, we continue downwards until we reach $x+y={\ell}+1$. Then we iterate the procedure  until we reach the bottom. 
For example, for ${\ell}=13$ and $\lambda=(10,10,9,6,5,4,4,3,1,1,1,1,0)$:

\begin{center}
$$
\begin{Tiles}{0.9}
\Dbloc{\Ttop\Tleft}\Dbloc{\Ttop}\Dbloc{\Ttop}\Dbloc{\Ttop}\Dbloc{\Ttop}\Dbloc{\Ttop}\Dbloc{\Ttop}\Dbloc{\Ttop}\Dbloc{\Ttop}\Dbloc{\Ttop\Tright}\Dbloc{\Ttopleftdot\Ttopdots}\Dbloc{\Ttopleftdot\Ttopdots}\Dbloc{\Ttopleftdot\Ttopdots}\Dbloc{\Ttopleftdot\Ttopdots\Tantidiagonal}\Dskip
\Dbloc{\Tleft}\Dspace\Dspace\Dspace\Dspace\Dbloc{}\Dspace\Dspace\Dspace\Dbloc{\Tright\Tbottom}\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dbloc{\Ddoubletext{c}{x+y=\ell+1}}\Dskip
\Dbloc{\Tleft}\Dspace\Dspace\Dspace\Dspace\Dbloc{}\Dbloc{\Tbottom}\Dbloc{\Tbottom}\Dbloc{\Tbottom\Tright}\Dbloc{\Trightdots\Tbottomrightdot}\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft}\Dspace\Dspace\Dspace\Dspace\Dbloc{\Tbottom\Tright}\Dbloc{\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots\Trightdots\Tbottomrightdot}\Dbloc{\Tantidiagonal\Tbottomleftdot}\Dskip
\Dbloc{\Tleft}\Dbloc{}\Dbloc{}\Dbloc{}\Dbloc{\Tbottom}\Dbloc{\Tleft}\Dspace\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft}\Dspace\Dspace\Dspace\Dbloc{\Tleft\Trightdots\Tbottomrightdot}\Dspace\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft}\Dspace\Dspace\Dspace\Dbloc{\Tleft\Trightdots\Tbottomrightdot}\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft}\Dbloc{\Tbottom}\Dbloc{\Tbottom\Tright}\Dbloc{\Ttop}\Dspace\Dbloc{\Tleftdots\Tbottomleftdot}\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft}\Dbloc{\Tleft\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots\Tbottomrightdot}\Dbloc{\Tleftdots\Tantidiagonal}\Dskip
\Dbloc{\Tleft\Tright}\Dspace\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft\Tright}\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft\Tbottom\Tright}\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tbottomleftdot\Trightdots\Tbottomrightdot\Tbottomdots}\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleftdots\Tantidiagonal}
\end{Tiles}
$$
\caption{}{\label{fig1}}
\end{center}


Let $n(\lambda)$ be the number of points of the dotted line on $x+y={\ell}+1$, which are not at the top or bottom. For example, we have $n((0,\dots,0))=0$, and for the ${\ell}$-partition $\lambda$ of Figure \ref{fig1}, we have $n(\lambda)=3$.

We shall describe the construction of this line in a more formal way. 

Let $k=n(\lambda)$. Set $i_n={\ell}+1$ for all $n>k$, $i_k=\lambda_1$, $i_{k-1}=\lambda_{{\ell}-i_k+2}$, $i_{k-2}=\lambda_{{\ell}-i_{k-1}+2}$, $\dots$, $i_{1}=\lambda_{{\ell}-i_2+2}$ and $i_p=0$ for all $p\leqslant 0$. We have $0<i_1<\dots <i_k<{\ell}+1$. The dotted line describes the shape of an ${\ell}$-partition 
\begin{equation}\label{partmax}
\lambda^M=(i_k^{{\ell}-i_k+1}, i_{k-1}^{i_k-i_{k-1}},\dots, i_1^{i_2-i_1}, 0^{i_1-1}).
\end{equation}

Any ${\ell}$-partition $\lambda$ whose associated dotted line gives the partition  $\lambda^M$ must necessarily contain the cells 
$$
(1,i_k),({\ell}-i_k+2,i_{k-1}),({\ell}-i_{k-1}+2,i_{k-2}),\dots, ({\ell}-i_2+2,i_1).
$$ 
The ``minimal'' ${\ell}$-partition in the sense of inclusion of diagrams that contains these cells is 
\begin{equation}\label{partmin}
\lambda^m=(i_k, i_{k-1}^{{\ell}-i_k+1},i_{k-2}^{i_k-i_{k-1}},\dots, i_1^{i_3-i_2}, 0^{i_2-2}).
\end{equation}

For example, take ${\ell}=13$ and $\lambda=(10,10,9,6,5,4,4,3,1,1,1,1,0)$, as above, we have $n(\lambda)=k=3$, $i_3=10,i_2=5,i_1=1$. The three distinguished cells above are  
$$
(1,10), (5,5), (10,1).
$$
So we have
$$
\begin{array}{c}
\lambda^M=(10,10,10,10, 5,5,5,5,5,1,1,1,1), \mbox{ and} \\
\lambda^m=(10,5,5,5,5,1,1,1,1,1,0,0,0).
\end{array}
$$
These partitions are illustrated in the figure below, where the distinguished cells are marked with $\times$, and $\lambda^M$ is the partition corresponding to the dotted line outside $\lambda$, while $\lambda^m$ is the one which corresponds to the dotted line inside $\lambda$. 
$$
\begin{Tiles}{1}\label{fig2}
\Dbloc{\Ttop\Tleft}\Dbloc{\Ttop}\Dbloc{\Ttop}\Dbloc{\Ttop}\Dbloc{\Ttop}\Dbloc{\Ttop\Tbottomleftdot\Tbottomdots}\Dbloc{\Ttop\Tbottomleftdot\Tbottomdots}\Dbloc{\Ttop\Tbottomleftdot\Tbottomdots}\Dbloc{\Ttop\Tbottomleftdot\Tbottomdots}\Dbloc{\Ttop\Tright\Tbottomleftdot\Tbottomrightdot\Tbottomdots\Dtext{c}{\times}}\Dbloc{\Ttopleftdot\Ttopdots}\Dbloc{\Ttopleftdot\Ttopdots}\Dbloc{\Ttopleftdot\Ttopdots}\Dbloc{\Ttopleftdot\Ttopdots\Ttoprightdot\Tantidiagonal}\Dskip
\Dbloc{\Tleft}\Dspace\Dspace\Dspace\Dspace\Dbloc{\Tleftdots\Tbottomleftdot}\Dspace\Dspace\Dspace\Dbloc{\Tright\Tbottom}\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dbloc{\Ddoubletext{c}{x+y=\ell+1}}\Dskip
\Dbloc{\Tleft}\Dspace\Dspace\Dspace\Dspace\Dbloc{\Tleftdots\Tbottomleftdot}\Dbloc{\Tbottom}\Dbloc{\Tbottom}\Dbloc{\Tbottom\Tright}\Dbloc{\Trightdots\Tbottomrightdot}\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft}\Dspace\Dspace\Dspace\Dspace\Dbloc{\Tleftdots\Tbottomleftdot\Tbottom\Tright}\Dbloc{\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots\Trightdots\Tbottomrightdot}\Dbloc{\Tantidiagonal\Tbottomleftdot}\Dskip
\Dbloc{\Tleft}\Dbloc{\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots\Tbottomrightdot}\Dbloc{\Tbottom\Dtext{c}{\times}}\Dbloc{\Tleft}\Dspace\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft\Trightdots\Tbottomrightdot}\Dspace\Dspace\Dspace\Dbloc{\Tleft\Trightdots\Tbottomrightdot}\Dspace\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft\Trightdots\Tbottomrightdot}\Dspace\Dspace\Dspace\Dbloc{\Tleft\Trightdots\Tbottomrightdot}\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft\Trightdots\Tbottomrightdot}\Dbloc{\Tbottom}\Dbloc{\Tbottom\Tright}\Dbloc{\Ttop}\Dspace\Dbloc{\Tleftdots\Tbottomleftdot}\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft}\Dbloc{\Tleft\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots\Tbottomrightdot}\Dbloc{\Tleftdots\Tantidiagonal}\Dskip
\Dbloc{\Tleft\Tbottomleftdot\Tbottomdots\Tbottomrightdot\Tright\Dtext{c}{\times}}\Dspace\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft\Tright}\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft\Tbottom\Tright}\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleftdots\Tbottomleftdot\Trightdots\Tbottomrightdot\Tbottomdots}\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleftdots\Tbottomleftdot\Tantidiagonal}
\end{Tiles}
$$

Observe that the difference $\lambda^M\setminus\lambda^m$ is a disjoint union of $k$ rectangles, denoted by $R_k,\dots,R_1$ from the top to the bottom. More precisely,
$$
R_j=\{(s,t);{\ell}-i_{p+1}+2<s<{\ell}-i_{p}+2 \mbox{ and } i_{p-1} <t \leqslant i_p\}.
$$
Inside each rectangle $R_j$, the shape of $\lambda$ could be described by a word $M_j$, whose letters are $d$ and $l$, where $d$ indicates a down step and $l$ indicates a left step. 

Let $h_j$ be the number of $d$'s in $M_j$, which is at most the height of $R_j$ and let $l_j$ be the number of $l$'s in $M_j$, which is the length of $R_j$. Then we have
$$
\begin{array}{c}
h_j=i_{j+1}-i_j-1 \mbox{ if } j\not=1, \mbox{ and } h_j\leqslant i_{j+1}-i_j-1 \mbox{ if } j=1, \\
l_j=i_j-i_{j-1},
\end{array}
$$
so $h_j\leqslant l_{j+1}-1$ and the equality holds if $j\not= 1$.
Furthermore the shape of $M_j$ is $l^{a_{j,0}}dl^{a_{j,1}}d \dots dl^{a_{j,h_j}}$, where $a_{j,i}\in\nset$, $0\leqslant i\leqslant h_j$. We then have that
\begin{equation}\label{equation_a_i}
l_j=\sum_{i=0}^{h_j} a_{j,i}.
\end{equation}
In the above example, we have $M_3=dldl^3dl$, $M_2=lddldl^2d$ and $M_1=ddl$.




We shall now generate a Dyck path step by step from the $M_j$. We call a peak of a Dyck path, an occurrence of $ud$ in the corresponding Dyck word. 

First, let $D_{k+1}$ be the Dyck path of length $2(\ell+1-i_k)$ containing $\ell+1-i_k$ peaks. Next, we have $M_k=l^{a_{k,0}}dl^{a_{k,1}}d \dots dl^{a_{k,h_k}}$. We insert $a_{k,0}$ peaks on the first peak of the already existing Dyck path $D_{k+1}$, then $a_{k,1}$ peaks on the second peak, and so on. We call $D_k$ the new Dyck path obtained. Observed that the highest peaks of $D_k$ are exactly those newly inserted, so there are exactly $l_k$. Since $h_{k-1}\leqslant l_k-1$, the procedure can then be iterated by inserting peaks only on highest peaks. Each intermediate Dyck path obtained after using the word $M_j$ is denoted by $D_{j}$. At the end, we obtain a Dyck path $D_{\lambda}$ of length $2\ell+2$.

For example, let us consider $\ell=7$ and $\lambda=(5,3,1,1,1,0,0)$:
$$
\begin{Tiles}{1}
\Dbloc{\Ttop\Tleft}\Dbloc{\Ttop\Tbottomleftdot\Tbottomdots}\Dbloc{\Ttop\Tbottomleftdot\Tbottomdots}\Dbloc{\Ttop\Tbottom}\Dbloc{\Ttop\Tbottom\Tright}\Dbloc{}\Dbloc{}\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft\Tbottomrightdot\Trightdots}\Dbloc{\Tbottom}\Dbloc{\Tright\Tbottom}\Dspace\Dspace\Dbloc{\Tleftdots\Tbottomleftdot}\Dbloc{\Tantidiagonal}\Dbloc{\Ddoubletext{c}{x+y=\ell+1}}\Dskip
\Dbloc{\Tleft\Tright}\Dbloc{\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots}
\Dbloc{\Tbottomleftdot\Tbottomdots\Tbottomrightdot\Trightdots}\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft\Tright\Tbottomdots\Tbottomrightdot\Tbottomleftdot}\Dspace\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft\Tright\Tbottom}\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleftdots\Tbottomleftdot\Trightdots\Tbottomrightdot}\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleftdots\Tbottomleftdot\Trightdots\Tbottomrightdot\Tbottomdots}\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tantidiagonal}\Dskip
\end{Tiles}
$$
\caption{}\label{example1}

We have $n(\lambda)=k=2$, $i_2=5$ and $i_1=1$. Then $D_3$ is the following Dyck path: 
\begin{center}
$$
\Gitter(7,3)(0,0)
\Koordinatenachsen(7,2)(0,0)
\Pfad(0,0),343434\endPfad
\Label\lu{0}(0,0)
\hskip4.5cm
$$
\end{center}

We have $M_2=l^2dl^2d$, so we first insert $2$ peaks on the first peak of $D_3$, then again two peaks on the second one. We obtain $D_2$:
\begin{center}
$$
\Gitter(15,4)(0,0)
\Koordinatenachsen(15,4)(0,0)
\Pfad(0,0),33434433434434\endPfad
\Label\lu{0}(0,0)
\hskip7.5cm
$$
\end{center}
Finally, $M_1=dl$ so we insert $a_{1,0}=0$ peak on the first highest peak of $D_2$ and $a_{1,1}=1$ peak on the second highest peak. We obtain $D_{\lambda}$:
\begin{center}
$$
\Gitter(18,5)(0,0)
\Koordinatenachsen(18
,5)(0,0)
\Pfad(0,0),3343344433434434\endPfad
\Label\lu{0}(0,0)
\Label\u{\scriptstyle 1}(1,0)
\Label\u{\scriptstyle 2}(2,0)
\Label\u{\scriptstyle 3}(3,0)
\Label\u{\scriptstyle 4}(4,0)
\Label\u{\scriptstyle 5}(5,0)
\Label\u{\scriptstyle 6}(6,0)
\Label\u{\scriptstyle 7}(7,0)
\Label\u{\scriptstyle 8}(8,0)
\Label\u{\scriptstyle 9}(9,0)
\Label\u{\scriptstyle 10}(10,0)
\Label\u{\scriptstyle 11}(11,0)
\Label\u{\scriptstyle 12}(12,0)
\Label\u{\scriptstyle 13}(13,0)
\Label\u{\scriptstyle 14}(14,0)
\Label\u{\scriptstyle 15}(15,0)
\Label\u{\scriptstyle 16}(16,0)
\Label\l{\scriptstyle 1}(0,1)
\Label\l{\scriptstyle 2}(0,2)
\Label\l{\scriptstyle 3}(0,3)
\hskip10.5cm
$$
\end{center}

By \cite{AKOP}, we have the following proposition.
\begin{prop}\label{map_D}
The map $D : \lambda \mapsto D_{\lambda}$ defines a bijection between the set of $\ell$-partitions and the set of Dyck paths of length $2\ell+2$.
\end{prop}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Dyck path and number of occurrences of ``$udu$''}
Let $\lambda=(\lambda_1, \dots, \lambda_{\ell})$ be an $\ell$-partition such that $n(\lambda)=k$. Let $D_{\lambda}$ be the Dyck path obtained from $\lambda$ as described in Section~\ref{section_AKOP_bijection}. We shall see how to count the number of occurrences of ``$udu$'' contained in $D_{\lambda}$.

A peak could be followed by a ``$u$'', a ``$d$'' or nothing in the Dyck word. If it is followed by a ``$u$'', we call it a $u$-peak. Each $u$-peak will give an ``$udu$'' and vice versa.

Let $1\leqslant j\leqslant k+1$. Let $u_j$ be the number of $u$-peaks in the Dyck path $D_j$. For example, $D_{k+1}$ contains $\ell-\lambda_1+1=\ell-i_k+1$ peaks, so it is easy to see that $u_{k+1}=\ell-\lambda_1$.

To construct $D_{j-1}$ from $D_j$, we add some peaks on the highest peaks of $D_j$. Then, one must understand how the insertion of $p$ peaks on a highest peak modifies the number of occurrences of ``$udu$''. Consider a peak $P$ of maximal height on a Dyck path. If we add  $p$ peaks, the part of the Dyck word which corresponds to $P$ (which was $ud$) becomes $uudud\dots udd$ (with $p$ $ud$), so we obtain $p-1$ 
occurrences of $udu$. If $P$ is a $u$-peak, then we also ``destroy'' the $udu$ given by $P$. So at the end, we only add $p-2$ occurrences of $udu$. For example,
let us consider the following Dyck path which contains $2$ occurrences of $udu$:
\begin{center}
$$
\Gitter(10,3)(0,0)
\Koordinatenachsen(10,3)(0,0)
\Pfad(0,0),34334344\endPfad
\Label\lu{0}(0,0)
\hskip4.5cm
$$
\caption{}\label{Dyckpath}
\end{center}
If we add $2$ peaks on the first highest peak, we add $2-2=0$ occurrences of $udu$. So we obtain the following Dyck path with still $2$ occurrences of $udu$:
\begin{center}
$$
\Gitter(13,4)(0,0)
\Koordinatenachsen(13,4)(0,0)
\Pfad(0,0),343334344344\endPfad
\Label\lu{0}(0,0)
\hskip5.5cm
$$
\end{center}

If $P$ is not a $u$-peak, then we do not ``destroy'' a $udu$, so we indeed add $p-1$ occurrences of ``$udu$''. For example, if we add $2$ peaks on the second highest peak of Figure \ref{Dyckpath}, we add $2-1=1$ occurrence of $udu$, so we obtain $3$ occurrences of $udu$ at the end:
\begin{center}
$$
\Gitter(13,4)(0,0)
\Koordinatenachsen(13,4)(0,0)
\Pfad(0,0),343343343444\endPfad
\Label\lu{0}(0,0)
\hskip5.5cm
$$
\end{center}


Set $a_{k+1,0}=\ell-i_k+1$, $M_{k+1}=l^{a_{k+1,0}}$, and $h_{k+1}=0$. We have seen that each word $M_j$ is in the form $l^{a_{j,0}}dl^{a_{j,1}}d \dots dl^{a_{j,h_j}}$. Let 
$$
\cala_j=\{(j,t); t\in\{0,\dots,h_j\}; a_{j,t}\not=0\},
$$ 
$$
\cala=\bigcup_{j=1}^k \cala_j.
$$ 
Recall from the construction that the number of highest peaks in $D_j$ is
\begin{equation}\label{equation_highest_peak}
\sum_{t=0}^{h_j} a_{j,i}=l_j.
\end{equation} 
Observe that a highest peak is a $u$-peak if it is not the last one of a consecutive group of highest peaks. Hence, the $q$-th peak of $D_j$ is not a $u$-peak if and only if there exists $r\in \{0,\dots, h_j\}$ such that $q=\sum_{s=0}^r a_{j,s}$. Set 
$$
\call_p=\left\{
(p,t); \mbox{there exists }\ 0\leqslant r\leqslant h_{p+1}; t+1=\sum_{q=0}^r a_{p+1,q}
\right\},
$$
$$
\begin{array}{ccc}
\calu_p=\cala_p\setminus \call_p, &
\displaystyle\call=\bigcup_{p=1}^k \call_p,& 
\displaystyle\calu=\bigcup_{p=1}^k \calu_p.
\end{array}
$$
Thus $\call_j$ corresponds exactly to the set of highest peaks in $D_j$ which are not $u$-peaks and where we insert new peaks. It follows that
$$
u_{j-1}=u_j +\sum_{(j-1,t)\in \calu_{j-1}}(a_{j-1,t}-2) +\sum_{(j-1,t)\in \call_{j-1}}(a_{j-1,t}-1).
$$

At the end of the construction, the number of occurrences of ``$udu$'' in $D_{\lambda}$ is $u_1$. By induction, we have  
$$
u_1= \ell-\lambda_1 +\displaystyle \sum_{(j,t)\in\calu}(a_{j,t}-2) + \sum_{(j,t)\in\call}(a_{j,t}-1).
$$
Since $\sum_{(j,t)\in\cala}a_{j,t}=\lambda_1$, we obtain the following proposition.
\begin{prop}\label{proposition_partition}
Let $\lambda$ be an $\ell$-partition. Then, the number of occurrences of ``$udu$'' in $D_{\lambda}$ is $\ell-2\sharp \calu-\sharp \call$.
\end{prop}


To illustrate this, we could follow again the construction of the Dyck path which corresponds to $\lambda=(5,3,1,1,1,0,0)$. We first have the Dyck path $D_3$ in Section~\ref{section_AKOP_bijection}, with $n-\lambda_1+1=3$ peaks, and $u_3=2$. Then we use the word $M_2=l^2dl^2d=l^{a_{2,0}}dl^{a_{2,1}}d$, where $a_{2,0}, a_{2,1}\in \call_2$, so we add $a_{2,0}-2+a_{2,1}-2=0$ peak. So $u_2=2$. Then we use the word $M_1=dl=l^{a_{1,0}}dl^{a_{1,1}}$, where $a_{1,1}\in \calu_1$, so we add $a_{1,1}-1=0$ peak. Hence, $u_1=2$.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Ad-nilpotent ideals of a parabolic subalgebra and Dyck paths}\label{ideaux_Dyck_path}

Let $I \subset \Pi$ and $\ilie$ be an ad-nilpotent ideal of
$\plie_I$.  We set 
$$
\Phi_{\ilie} =  \{ \alpha \in \Delta^+ \setminus \Delta_I;\ 
\mathfrak{g}_{\alpha } \subseteq \ilie \}.
$$ 
Then $\ilie = 
\bigoplus_{\alpha \in \Phi_{\ilie} } \mathfrak{g}_{\alpha}$ and if 
$\alpha \in \Phi_{\ilie}$, $\beta \in \Delta^+\cup \Delta_I$ are such that $\alpha +\beta
\in \Delta^+$, then $\alpha +\beta \in \Phi_{\ilie}$. 

Conversely, set 
$$ 
\calf_I = \{ \Phi \subset \Delta^+\setminus \Delta_I;\mbox{if } \alpha \in \Phi,
\beta \in \Delta^+\cup \Delta_I, \alpha +\beta \in \Delta^+, 
\mbox{then} \ \alpha +\beta\in \Phi \}.
$$
Then for $\Phi \in \calf_I$, $\ilie_{\Phi} = \bigoplus_{\alpha \in
\Phi} \mathfrak{g}_{\alpha}$ is an ad-nilpotent ideal of $\plie_I$. 

We obtain therefore a bijection
$$
\{\mbox{ad-nilpotent ideals of } \plie_I \}  \rightarrow  \calf_I ,\ 
\ilie  \mapsto  \Phi_{\ilie}.
$$


Recall the following partial order on $\Delta^+$: 
$\alpha < \beta$ if $\beta -\alpha$ is a 
sum of positive roots. Then it is easy to see that $\Phi \in
\calf_{\emptyset}$ 
if and only if for all $\alpha \in \Phi, \beta \in \Delta^+$, such
that $\alpha < \beta$, then 
$\beta \in \Phi$. 

Let $\Phi\in\calf_{\emptyset}$. Set
$$
\Phi_{min}=\{\beta\in\Phi;\beta-\alpha\not\in\Phi, \mbox{ for all } \alpha\in \Delta^+\}.
$$
Then, $\Phi_{min}$ is an antichain of $\Delta^+$ with respect to the above partial order. Conversely, if we consider an antichain $\Gamma$, then, the set of roots which are bigger than any one of the elements of $\Gamma$ is an element of $\calf_{\emptyset}$.



As in \cite{CP}, we display the positive 
roots $\Delta^+$ in the Ferrers diagram $T_{\ell}$ of $({\ell},{\ell}-1,\dots,1)$ as follows: we assign to each box in the $i$-th
row and the $j$-th column, labelled $(i,j)$ in $T_{\ell}$, 
a positive root $t_{i,j}=\alpha_i+\cdots +\alpha_{{\ell}-j+1}$, $1\leqslant i,
j\leqslant {\ell}$. 

For example, for ${\ell}=5$, we have
$$
\overbrace{
\vbox{\offinterlineskip
\halign{#&#&#&#&#&#\cr
\htrait&\htrait&\htrait&\htrait&\htrait\cr
\sboite{$t_{1,1}$}&\sboite{$t_{1,2}$}&\sboite{$t_{1,3}$}&\sboite{$t_{1,4}$}&%
\sboite{$t_{1,5}$}&\vtrait\cr
\htrait&\htrait&\htrait&\htrait&\htrait\cr
\sboite{$t_{2,1}$}&\sboite{$t_{2,2}$}&\sboite{$t_{2,3}$}%
&\sboite{$t_{2,4}$}&\vtrait\cr
\htrait&\htrait&\htrait&\htrait\cr
\sboite{$t_{3,1}$}&\sboite{$t_{3,2}$}&\sboite{$t_{3,3}$}&\vtrait\cr
\htrait&\htrait&\htrait\cr
\sboite{$t_{4,1}$}&\sboite{$t_{4,2}$}&\vtrait\cr
\htrait&\htrait\cr
\sboite{$t_{5,1}$}&\vtrait\cr
\htrait\cr
}
}
}^{\ell}
$$

Observe that given two positive roots $\alpha$ and $\beta$, $\alpha$ is bigger than or equal to $\beta$ if the box corresponding to $\alpha$ is in the quadrant north-west of the box corresponding to $\beta$. 
It follows easily that the map which sends an element $\Phi\in\calf_{\emptyset}$ to the subdiagram of $T_{\ell}$ consisting of the boxes corresponding to the roots of $\Phi$ defines a bijection between $\calf_{\emptyset}$ and the set of northwest flushed 
subdiagrams of $T_{\ell}$, i.e with the set of subdiagrams which contain the quadrant north-west of their boxes. Hence, by Section~\ref{partition_Dyck_path}, we obtain a bijection $\sigma$ from $\calf_{\emptyset}$ to the set of ${\ell}$-partitions.

By Proposition~\ref{map_D}, $D\circ\sigma$ is a bijection from $\calf_{\emptyset}$ to the set of Dyck paths of length $2{\ell}+2$.


For $\Phi\in\calf_{\emptyset}$, set 
$$
I_{\Phi}=\{\alpha\in\Pi;\Phi\in\calf_{\{\alpha\}}\}.
$$
It is the maximal element of $\{I\subset\Pi; \Phi\in\calf_I\}$ with respect to inclusion order.  We shall see how to link the number of occurrences of ``$udu$'' of the Dyck path 
$(D\circ\sigma)(\Phi)$ and the cardinality of $I_{\Phi}$.


Set $\alpha_{i,j}=\alpha_i +\dots+\alpha_j$, for all $1\leqslant i\leqslant j \leqslant \ell$. We have easily the following lemma.
\begin{lemme}\label{lemme_Phi_min}
Let $I\subset\Pi$. An element $\Phi\in\calf_{\emptyset}$ is an element of $\calf_I$ if and only if for all $\alpha_{i,j}\in\Phi_{min}$, we have $\alpha_i,\alpha_j\not\in I$.
\end{lemme}
It follows from Lemma~\ref{lemme_Phi_min} that
$$
I_{\Phi}=\Pi\setminus\{\alpha_i\in\Pi;\mbox{there exists } \alpha_{i,j} \mbox{ or } \alpha_{k,i}\in \Phi_{min}\}.
$$

The problem is not to count the same root twice. For example, in $A_7$, for $\Phi_{min}=\{\alpha_{1,3},\alpha_{2,5},\alpha_{5,7}\}$, we have $\Pi\setminus I_{\Phi}=\{\alpha_1,\alpha_2,\alpha_3,\alpha_5,\alpha_7\}$ but we find $\alpha_5$ in the beginning or in the end of the support of two roots in $\Phi_{min}$. So if we set
$$
L=\{\alpha_{i,j}\in\Phi_{min};\mbox{ there exists a root of shape } \alpha_{p,i}\in\Phi_{min}\},
$$
$$
U=\Phi_{min}\setminus L,
$$
we obtain that 
\begin{equation}\label{cardinal_I_max}
\sharp I_{\Phi}=l-2\sharp U -\sharp L.
\end{equation}

Let $\lambda=\sigma(\Phi)$, $F$ its Ferrers diagram and $D_{\lambda}=D(\lambda)$ be the Dyck path which corresponds to $\lambda$ via the AKOP-bijection. Let $\alpha_{i,j}\in\Phi_{min}$. Then the cell $(i,\ell+1-j)=(i,\lambda_i)$ of $\alpha_{i,j}$ in $F$ is a south-east corner of the diagram and two cases are possible: there exists a rectangle $R_{p}$ such that $(i,\lambda_i)\in R_{p}$ or $(i,\lambda_i)$ is not in any rectangle. If the latter case occurs, then $(i,\ell+1-j)$ is above a rectangle $R_p$. For example, if $\lambda=(5,3,1,1,1,0,0)$, we have that $\alpha_{2,5},\alpha_{5,7}$ are in the first case and $\alpha_{1,3}$ is in the second case.
$$
\begin{Tiles}{1.1}
\Dbloc{\Ttop\Tleft}\Dbloc{\Ttop\Tbottomleftdot\Tbottomdots}\Dbloc{\Ttop\Tbottomleftdot\Tbottomdots}\Dbloc{\Ttop\Tbottom}\Dbloc{\Ttop\Tbottom\Tright\Dtext{c}{\alpha_{1,3}}}\Dbloc{}\Dbloc{}\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft\Tbottomrightdot\Trightdots}\Dbloc{\Tbottom}\Dbloc{\Tright\Tbottom\Dtext{c}{\alpha_{2,5}}}\Dspace\Dspace\Dbloc{\Tleftdots\Tbottomleftdot}\Dbloc{\Tantidiagonal}\Dbloc{\Ddoubletext{c}{x+y=\ell+1}}\Dskip
\Dbloc{\Tleft\Tright}\Dbloc{\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots}\Dbloc{\Tbottomleftdot\Tbottomdots\Dtext{c}{R_2}}
\Dbloc{\Tbottomleftdot\Tbottomdots\Tbottomrightdot\Trightdots}\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft\Tright\Tbottomdots\Tbottomrightdot\Tbottomleftdot}\Dspace\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleft\Tright\Tbottom\Dtext{c}{\alpha_{5,7}}}\Dspace\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleftdots\Tbottomleftdot\Trightdots\Tbottomrightdot}\Dspace\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tleftdots\Tbottomleftdot\Trightdots\Tbottomrightdot\Tbottomdots\Dtext{c}{R_1}}\Dbloc{\Tantidiagonal}\Dskip
\Dbloc{\Tantidiagonal}\Dskip
\end{Tiles}
$$
%\caption{}\label{example_precis}


If $\alpha_{i,j}$ is in the rectangle $R_{p}$, then the cell $(i,\lambda_i)=(i,{\ell}-j+1)$ which corresponds to $\alpha_{i,j}$ in $F$ satisfies
\begin{equation}\label{equation_i}
{\ell}-i_{p+1}+2<i<{\ell}-i_p+2,
\end{equation}
\begin{equation}\label{equation_lambda}
i_{p-1}<\lambda_i\leqslant i_p,
\end{equation}
and so we have
\begin{equation}\label{equation_j}
{\ell}-i_{p}+1\leqslant j< {\ell}-i_{p-1}+1.
\end{equation}

If $\alpha_{i,j}$ is above the rectangle $R_{p}$, then the cell $(i,{\ell}-j+1)$ which corresponds to $\alpha_{i,j}$ in $F$ satisfies
\begin{equation}\label{equation_upper}
(i,{\ell}-j+1)=({\ell}-i_{p+1}+2,i_p).
\end{equation}

Define the map $r$ from $\Phi_{min}$ to $\{1,\dots,k\}$ which associates to $\alpha_{i,j}$ the integer $r(\alpha_{i,j})=p$ such that $\alpha_{i,j}$ is in or immediately above the rectangle $R_{p}$. 


Let $\alpha_{i,j}\in\Phi_{min}$ and $p=r(\alpha_{i,j})$. Since the cell $(i,{\ell}-j+1)$ which contains $\alpha_{i,j}$ in $T_{\ell}$ is a south-east corner, there is a horizontal line under this cell. If 
$c=(i,{\ell}-j+1)$ is in the rectangle $R_p$, then it is at the row $q=i-({\ell}-i_{p+1}+2)$ of $R_p$ and the line under $c$ correspond to the part $l^{a_{p,q}}$ in $M_p$. Furthermore $(p,q)\in\cala_p$.

If $c$ is immediately above the rectangle $R_p$, then the line under $c$ corresponds to $l^{a_{p,0}}$ in $M_p$ and $(p,0)\in\cala_p$. Since in this case, by \eqref{equation_upper} we have $(i,\ell-j+1)=(\ell-i_{p+1}+2,i_p)$, we obtain that $i-(\ell-i_{p+1}+2)=0$. We can define in any case the map $s$ from $\Phi_{min}$ to $\nset$ by 
\begin{equation}\label{equation_q}
s(\alpha_{i,j})=i-(\ell-i_{r(\alpha_{i,j})+1}+2).
\end{equation}
Furthermore, in both cases, the line under the cell which contains $\alpha_{i,j}$ is the part $l^{a_{r(\alpha_{i,j}),s(\alpha_{i,j})}}$ in $M_{r(\alpha_{i,j})}$ and $(r(\alpha_{i,j}),s(\alpha_{i,j}))\in\cala_{r(\alpha_{i,j})}$.

Conversely, let $(p,q)\in\cala_p$. Then, there is a horizontal line under the row $i=q-\ell-i_{p+1}+2$ of $F$ which is under a south-east corner of $F$. This south-east corner is a cell $(i,\lambda_i)$ which corresponds to a root $\alpha_{i,j}$, where $\ell-j+1=\lambda_i$. So we have a bijection
$$
\begin{array}{ccrcl}
\Psi &:&\Phi_{min}&\rightarrow& \cala \\
&&\alpha_{i,j}&\mapsto & (r(\alpha_{i,j}),s(\alpha_{i,j})).
\end{array}
$$


\begin{lemme}
We have $\Psi(U)=\calu$ and $\Psi(L)=\call$. 
\end{lemme}

\begin{proof}
Since $L=\Phi_{min}\setminus U$ and $\call=\cala\setminus\calu$, it suffices to prove that $\Psi(L)=\call$. 

Let $\alpha_{i,j}\in L$. Set $p=r(\alpha_{i,j})$, $q=s(\alpha_{i,j})$ and let $c=(i,\lambda_i)$ be the cell which corresponds to $\alpha_{i,j}$ in $F$. 

First assume that $i=j$. Then, we have  $c=(i,{\ell}-i+1)$. If $c\in R_{p}$, then by \eqref{equation_i} and \eqref{equation_j}, we have
$$
i={\ell}-i_p+1,
$$
so by \eqref{equation_q}, we have that $q=i_{p+1}-i_p-1$ so by \eqref{equation_a_i}, $a_{p,q}\in\call_{p}$.

If $c$ is above $R_{p}$, then by \eqref{equation_upper}, we have $c=(i,{\ell}-i+1)=({\ell}-i_{p+1}+2,i_p)$, so $q=0$ and $i_{p+1}-i_p=1$, hence by \eqref{equation_a_i} we also have $a_{p,q}\in\call_{p}$.

Now assume that $i\not=j$ and there exists a root of shape $\alpha_{m,i}\in\Phi_{min}$. Set $t=r(\alpha_{m,i})$. Let $(m,\lambda_m)=(m,{\ell}-i+1)$ be the cell which corresponds to $\alpha_{m,i}$ in $\lambda$. If $c\in R_{p}$, then by \eqref{equation_i}, we have 
$$
i_p\leqslant \lambda_m\leqslant i_{p+1}-2.
$$
So either $(m,\lambda_m)\in R_{p+1}$ or $(m,\lambda_m)=({\ell}-i_{p+1}+2,i_p)$.

If $(m,\lambda_m)\in R_{p+1}$, then between the columns $i_{p+1}$ and $\lambda_m={\ell}-i+1$, we have $i_{p+1}-({\ell}-i+1)$ columns, so there exists $n$ such that $\sum_{u=0}^n a_{p+1,u}=i_{p+1}-({\ell}-i+1)$. Furthermore, by \eqref{equation_q}, we have $q=i-({\ell}-i_{p+1}+2)$, hence $a_{p,q}\in\call_{p}$.

If $(m,\lambda_m)=({\ell}-i_{p+1}+2,i_p)$, then $i={\ell}-i_p+1$ and by \eqref{equation_q}, we have that
$$
q=({\ell}-i_p+1)-({\ell}-i_{p+1}+2)=i_{p+1}-i_p-1.
$$
Hence, by \eqref{equation_a_i}, we have $a_{p,q}\in\call_{p}$.

Conversely, let $a_{p,q}\in \call_p$, then there exists $0\leqslant t\leqslant h_{p+1}$ such that $q+1=\sum_{f=0}^t a_{p+1,f}$. There also exists $\alpha_{i,j}\in\Phi_{min}$ such that $r(\alpha_{i,j})=p$ and $s(\alpha_{i,j})=q$. By \eqref{equation_q}, we have that
$$
q=i-({\ell}-i_{p+1}+2).
$$
Observe that for all $0\leqslant j\leqslant h_{p+1}$, there exists a south-east corner $(n_j,\lambda_{n_j})$ in or above the rectangle $R_{p+1}$ such that
$$
\lambda_{n_j}=i_{p+1}-\sum_{f=0}^j a_{p+1,f}.
$$
So there exists a south-east corner $(n_j,\lambda_{n_j})$ such that
$$
\lambda_{n_j}=i_{p+1}-(q+1)={\ell}-i+1.
$$
The element of $\Phi_{min}$ which corresponds to the cell $(n_j,\lambda_{n_j})$ is $\alpha_{n_j,i}$, so we have $\alpha_{i,j}\in L$.
\end{proof}

It follows by Proposition~\ref{proposition_partition} and Equation~\eqref{cardinal_I_max} that we have the following theorem.
\begin{theoreme}
There is a bijection between the elements $\Phi\in\calf_{\emptyset}$ such that $\sharp I_{\Phi}=r$ and the Dyck paths of length $2{\ell}+2$ having $r$  occurrences of ``$udu$''.
\end{theoreme}

Since the number of Dyck paths having a fixed number of occurrences of $udu$ is calculated in Theorem~2.1 of \cite{Su}, we have the following corollary.
\begin{corollaire}\label{I_max_A}
The number of elements of $\Phi\in\calf_{\emptyset}$ such that $\sharp I_{\Phi}=r$ is
$$
\left(\begin{array}{c}
{\ell}\\
r
\end{array}
\right)
\sum_{k=0}^{[{\ell}-r/2]} \left(\begin{array}{c}
{\ell}-r\\
2k
\end{array}
\right)
\calc_{k}
$$
where $\calc_{k}$ denotes the $k$-th Catalan number.
\end{corollaire}

\begin{exemple}
Let $N_r^{\ell}$ be the number of elements $\Phi\in\calf_{\emptyset}$ such that $\sharp I_{\Phi}=r$. We have by Corollary~\ref{I_max_A}:
$$
\begin{array}{|c||c|c|c|c|c|}
\hline
r &N_r^{1}&N_r^{2}&N_r^{3} &N_r^{4}&N_r^{5}\\
\hline
0 &1 &2 &4 &9 &21\\
\hline
1  &1 &2 &6 &16 &45 \\
\hline
2&  &1 &3 &12 &40\\
\hline
3&& &1& 4& 20\\
\hline
4&&&& 1& 5\\
\hline
5& & & &&1 \\
\hline
\end{array}
$$
\end{exemple}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Duality}\label{section_dualite}
We shall construct a duality between the elements of $\calf_{\emptyset}$ such that $\sharp \Phi_{min}=p$ and those such that $\sharp \Phi_{min}={\ell}-p$.

\begin{prop}\label{prop_phi_min1}
Let $\Phi\in\calf_{\emptyset}$. Let $N$ be the number of peaks in $(D\circ\sigma)(\Phi)$, then we have
$$
\sharp \Phi_{min}={\ell}-(N-1).
$$
\end{prop}


\begin{proof}
Let $\lambda=\sigma(\Phi)$ be the corresponding ${\ell}$-partition. Recall that the construction of $D(\lambda)$ is iterative. At each step, when we add $a_{p,q}$ peaks to a 
highest peak, for $(p,q)\in\cala_p$, we also ``destroy'' the initial highest peak. So, we add only $a_{p,q}-1$ peaks. At the end of the construction we have
$$
{\ell}-\lambda_1+1+\sum_{p=1}^k \sum_{(p,q)\in\cala_p}(a_{p,q}-1)
$$
peaks. Since $\sum_{p=1}^k \sum_{(p,q)\in\cala_p}a_{p,q}=\sum_{(p,q)\in\cala} a_{p,q}=\lambda_1$ and $\cala$ is in bijection with $\Phi_{min}$ by Section~\ref{ideaux_Dyck_path}, we obtain the result.
\end{proof}

\begin{prop}\label{prop_phi_min2}
Let $\Phi\in\calf_{\emptyset}$ and $p$ be the number of peaks in $(P\circ\sigma)(\Phi)$, then we have
$$
\sharp \Phi_{min}=p-1.
$$
\end{prop}

\begin{proof}
The result is clear by the construction of $(P\circ\sigma)(\Phi)$ defined in Section~\ref{partition_Dyck_path}.
\end{proof}

\begin{theoreme}
The map $\sigma^{-1}\circ P^{-1}\circ D\circ\sigma$ induces a bijection from $\calf_{\emptyset}$ to $\calf_{\emptyset}$ which sends 
$\Phi\in\calf_{\emptyset}$ such that $\sharp \Phi_{min}=p$ to $\Psi\in\calf_{\emptyset}$ such that $\sharp \Psi_{min}={\ell}-p$.
\end{theoreme}

For example, in $sl_4(\cset)$, the element $\Phi=\{\theta\}\in\calf_{\emptyset}$ corresponds to the partition $\lambda=(1,0,0)$, and the Dyck path $D_{\lambda}$ is:
\begin{center}
$$
\Gitter(9,4)(0,0)
\Koordinatenachsen(9,4)(0,0)
\Pfad(0,0),33443434\endPfad
\Label\lu{0}(0,0)
\hskip4.5cm
$$
\end{center}
Then, $P^{-1}(D_{\lambda})=(3,2,0)$ which is the partition which corresponds to $\Psi$ such that $\Psi_{min}=\{\alpha_1, \alpha_2\}$.

\begin{remarque}
It was  proved in \cite{P} that when $\glie$ is a simple Lie algebra of type $A$ or $C$, the number of elements 
$\Phi\in\calf_{\emptyset}$ such that 
$\sharp\Phi_{min}=p$ is the same as the number of elements $\Phi\in\calf_{\emptyset}$ such that $\sharp\Phi_{min}=\ell-p$. 
But the duality of \cite{P} is not the same as the one defined above. For example, in $sl_4(\cset)$, if we consider $\Phi=\{\theta\}$ like above, the dual ideal defined by \cite{P} is $\Psi$ where $\Psi_{min}=\{\alpha_1+\alpha_2, \alpha_3\}$.


\end{remarque}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{AKOP}

\bibitem[AKOP]{AKOP}\textsc{G. E. Andrews, C. Krattenthaler, L. Orsina and P. Papi.} \emph{Ad-nilpotent $\blie$-ideals in $sl(n)$ having a fixed class of nilpotence: combinatorics and enumeration.} Trans. Amer. Math. Soc. {\bf 354} (2002), 3835--3853.

\bibitem[CP]{CP} \textsc{P. Cellini, P. Papi}. \emph{Ad-nilpotent
  ideals of a Borel subalgebra}. J. Algebra {\bf 225} (2000), 130--140.


\bibitem[Pa]{P} \textsc{D.I. Panyushev}. \emph{Ad-nilpotent ideals of
  a Borel subalgebra: generators and duality}. J. Algebra {\bf 274} (2004),
  822--846.

\bibitem[R]{R} \textsc{C. Righi}. \emph{Ad-nilpotent ideals of a parabolic subalgebra}, J. Algebra {\bf 319} (2008), 1555--1584.


\bibitem[Sun]{Su} \textsc{Y. Sun}. \emph{The statistic "number of udu's" in Dyck paths.}
Discrete Math. {\bf 287} (2004), 177--186.


\end{thebibliography}
\end{document}


