% $Date: 2004/10/04 17:25:12 $ $Author: nthiery $ $Revision: 1.1.1.2 $

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

  \begin{MInfo}
    \eng{table of the Pe-basis}
  \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 elements of the \Mex{Pe}-basis,
which is the dual basis of the basis of (increasing) monomials (less than
the ``staircase'' monomial), 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)}.
 
When called with the second argument \Mex{X}, it returns the table of
all  polynomials in \Mmath{Sn}{S_n}, expressed in the Schubert basis.
}
\end{MIntro}
%
\begin{MExample}{ex1}
%
\begin{Mexin}
t:=muEC::SP::TablePe(3);
\end{Mexin}
\begin{Mexout}
                  table(
                                  2
                    [2, 1, 0] = x1  x2,
                                  2
                    [1, 1, 0] = x1  + x2 x1,
                    [1, 0, 0] = x1 + x2,
                    [2, 0, 0] = x1 x2,
                    [0, 1, 0] = x1,
                    [0, 0, 0] = 1
                  )
\end{Mexout}
\begin{Mexin}
muEC::SP::TablePe(3, X);
\end{Mexin}
\begin{Mexout}
            table(
              [2, 1, 0] = X[3, 2, 1],
              [1, 1, 0] = X[2, 3, 1] + X[3, 1, 2],
              [1, 0, 0] = X[1, 3, 2],
              [2, 0, 0] = X[2, 3, 1],
              [0, 1, 0] = X[2, 1],
              [0, 0, 0] = X[1]
            )
\end{Mexout}
\end{MExample}
%
\MRelatedFunctions{TableX,TableXX}
\end{Mhelppage}
