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

\begin{Mhelppage}{TableYY}{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 codes of permutations in \Mmath{Sn}{S_n}.
The polynomials are expressed in the basis of monomials.

When called with the second argument perm, the second alphabet (the \Mex{y}i's)
is specialized as the  permutation \Mex{perm} of the first one (the \Mex{xi}'s):
in other words, \Mex{y.i} is sent to \Mex{x.perm[i]}.
}
\end{MIntro}
%
\begin{MExample}{ex1}
%
\begin{Mexin}
t:=muEC::SP::TableYY(3);
\end{Mexin}
\begin{Mexout}
         table(
           [2, 1, 0] = (x1 - y1) (x1 - y2) (x2 - y1),
           [1, 1, 0] = (x1 - y1) (x2 - y1),
           [1, 0, 0] = x1 - y1,
           [2, 0, 0] = (x1 - y1) (x1 - y2),
           [0, 1, 0] = x1 + x2 - y1 - y2,
           [0, 0, 0] = 1
         )
\end{Mexout}
%
\begin{Mexin}
t[ [0,1,0] ];
\end{Mexin}
\begin{Mexout}
                       x1 + x2 - y1 - y2
\end{Mexout}
%
\begin{Mexin}
muEC::SP::TableYY(3, [3,2,1]);
\end{Mexin}
\begin{Mexout}
         table(
           [2, 1, 0] = (x1 - x2) (x1 - x3) (x2 - x3),
           [1, 1, 0] = (x1 - x3) (x2 - x3),
           [1, 0, 0] = x1 - x3,
           [2, 0, 0] = (x1 - x2) (x1 - x3),
           [0, 1, 0] = x1 - x3,
           [0, 0, 0] = 1
         )
\end{Mexout}
\end{MExample}
%
\MRelatedFunctions{TableY}
\end{Mhelppage}
