% $Date: 2003/10/28 21:59:21 $ $Author: nthiery $ $Revision: 1.1.1.2 $

\begin{Mhelppage}{TableY}{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 codes of 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 Sn, 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::TableY(3);
\end{Mexin}
\begin{Mexout}
                    table(
                                    2
                      [2, 1, 0] = x1  x2,
                      [1, 1, 0] = x1 x2,
                      [1, 0, 0] = x1,
                                    2
                      [2, 0, 0] = x1 ,
                      [0, 1, 0] = x1 + x2,
                      [0, 0, 0] = 1
                    )
\end{Mexout}
\begin{Mexin}
t[ [2,1,0] ];
\end{Mexin}
\begin{Mexout}
                              2
                            x1  x2
\end{Mexout}

\begin{Mexin}
muEC::SP::TableY(3, Pe);
\end{Mexin}
\begin{Mexout}
           table(
             [2, 1, 0] = Pe[2, 1, 0],
             [1, 1, 0] = Pe[2, 0, 0],
             [1, 0, 0] = Pe[1, 0],
             [2, 0, 0] = Pe[1, 1, 0] - Pe[2, 0, 0],
             [0, 1, 0] = Pe[1, 0, 0],
             [0, 0, 0] = Pe[0]
           )
\end{Mexout}
\end{MExample}
%
\MRelatedFunctions{TableYY}
\end{Mhelppage}
