% $Date: 2003/09/15 16:37:45 $ $Author: nthiery $ $Revision: 1.1.1.1 $

\begin{Mhelppage}{TableX}{S. Veigneau}

  \begin{MInfo}
    \eng{table of all Schubert polynomials}
  \end{MInfo}
      
\begin{MCalls}
 \Mcall{n}
\end{MCalls}
%
\begin{MParameters}
  \param{n}
  \eng{any positive integer denoting the degree of a symmetric group} 
\end{MParameters}
%
\begin{MIntro}
  \eng{
The {\Mthis} function returns the table of all Schubert polynomials
indexed by permutations in \Mmath{Sn}{S_n}.
The  polynomials are expressed in the basis of monomials.
 
When called with the second argument \Mex{Pe}, it returns the table of
all Schubert polynomials in \Mmath{Sn}{S_n}, expressed in the dual basis of the
monomial basis, i.e. the basis of products of elementary symmetric functions
on an alphabet flag.
For instance, \Mex{Pe[1, 2, 0, 1, 0]} stands for the product
\Mex{e1(x1,x2,x3,x4)*e2(x1,x2,x3)*e1(x1)}.
}
\end{MIntro}
%
\begin{MExample}{ex1}
%
\begin{Mexin}
t:=muEC::SP::TableX(3);
\end{Mexin}
\begin{Mexout}
                    table(
                                    2
                      [3, 2, 1] = x1  x2,
                      [2, 3, 1] = x1 x2,
                      [2, 1, 3] = x1,
                                    2
                      [3, 1, 2] = x1 ,
                      [1, 3, 2] = x1 + x2,
                      [1, 2, 3] = 1
                    )
\end{Mexout}
% \begin{Mexout}[hide]
% table([3, 2, 1]=x1^2*x2,[2, 3, 1]=x1*x2 ,[2, 1, 3]=x1,[3, 1, 2]=x1^2,
% [1, 3, 2]=x1 + x2,[1, 2, 3]= 1)
% \end{Mexout}
%
\begin{Mexin}
t[ [2,3,1] ];
\end{Mexin}
\begin{Mexout}
                             x1 x2
\end{Mexout}
% \begin{Mexout}[hide]
                            % x1*x2
% \end{Mexout}
%
\begin{Mexin}
muEC::SP::TableX(3, Pe);
\end{Mexin}
\begin{Mexout}
           table(
             [3, 2, 1] = Pe[2, 1, 0],
             [2, 3, 1] = Pe[2, 0, 0],
             [2, 1, 3] = Pe[1, 0],
             [3, 1, 2] = Pe[1, 1, 0] - Pe[2, 0, 0],
             [1, 3, 2] = Pe[1, 0, 0],
             [1, 2, 3] = Pe[0]
           )
\end{Mexout}
\end{MExample}
%
\MRelatedFunctions{TableXX}
\end{Mhelppage}
