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

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

  \begin{MInfo}
    \eng{table of all double 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}
  \param{perm}
  \eng{any list denoting a permutation} 
\end{MParameters}
%
\begin{MIntro}
  \eng{
The {\Mthis} function returns the table of all double Schubert polynomials
indexed by permutations in \Mmath{Sn}{S_n}. The polynomials are expressed
in the basis of monomials.

When the function is called with the second  argument  perm, the  second
alphabet (the yi's) is specialized as the permutation perm of the  first
one (the xi's): in other words, y.i is sent to x.perm[i].
}
\end{MIntro}
%
\begin{MExample}{ex1}
%
\begin{Mexin}
t:=muEC::SP::TableXX(3);
\end{Mexin}
\begin{Mexout}
         table(
           [3, 2, 1] = (x1 - y1) (x1 - y2) (x2 - y1),
           [2, 3, 1] = (x1 - y1) (x2 - y1),
           [2, 1, 3] = x1 - y1,
           [3, 1, 2] = (x1 - y1) (x1 - y2),
           [1, 3, 2] = x1 + x2 - y1 - y2,
           [1, 2, 3] = 1
         )
\end{Mexout}
% \begin{Mexout}[hide]
% table([3, 2, 1]=(x1 + y1*(-1))*(x1 + y2 *(-1))*(x2 + y1*(-1)),
% [2, 3, 1]=(x1 + y1*(-1))*(x2 + y1*(-1)),[2, 1, 3]=x1 + y1*(-1),
% [3, 1, 2]=(x1 + y1*(-1))*(x1 + y2*(-1)),[1, 3, 2]=x1 + x2 + y1*(-1) + y2*(-1),
% [1, 2, 3]=1)
% \end{Mexout}
%
\begin{Mexin}
t[ [3,1,2] ];
\end{Mexin}
\begin{Mexout}
                      (x1 - y1) (x1 - y2)
\end{Mexout}
% \begin{Mexout}[hide]
                % (x1 + y1*(-1))*(x1 + y2*(-1))
% \end{Mexout}
%
\begin{Mexin}
muEC::SP::TableXX(3, [3,2,1]);
\end{Mexin}
\begin{Mexout}
         table(
           [3, 2, 1] = (x1 - x2) (x1 - x3) (x2 - x3),
           [2, 3, 1] = (x1 - x3) (x2 - x3),
           [2, 1, 3] = x1 - x3,
           [3, 1, 2] = (x1 - x2) (x1 - x3),
           [1, 3, 2] = x1 - x3,
           [1, 2, 3] = 1
         )
\end{Mexout}
% \begin{Mexout}[hide]
% table([3, 2, 1]=(x1 + x2*(-1))*(x1 + x3 *(-1))*(x2 + x3*(-1)),
% [2, 3, 1]=(x1 + x3*(-1))*(x2 + x3*(-1)),[2, 1, 3]=x1 + x3*(-1),
% [3, 1, 2]=(x1 + x2*(-1))*(x1 + x3*(-1)),[1, 3, 2]=x1 + x3*(-1),
% [1, 2, 3]=1)
% \end{Mexout}
\end{MExample}

%
\MRelatedFunctions{TableX}
\end{Mhelppage}
