% $Id: ribbonsTableaux.tex,v 1.1.1.3 2005/01/26 13:42:24 nthiery Exp $
\begin{Mhelppage}{ribbonsTableaux}{F.~Descouens}
  [ImplState=stable,TestState=stable,DocState=beta]
%\Mproofread{...}{...}{...}

\begin{MInfo}
  \eng{Ribbon Tableaux}
\end{MInfo}

\begin{MIntro}
  \eng{The library \Mthis{} provides functions for generating and manipulating ribbon tableaux and their spin and cospin statistics. These objects are typed objects (as \Mup{combinat::trees}, \Mup{combinat::skewTableaux} and \Mup{combinat::ribbons}).}
\end{MIntro}

\MKeywords{combinatorics,tableaux,skewTableaux,ribbons,descents of a permutation,partitions}

\begin{MDetails}
  
  \eng{A $k$-ribbon tableaux \Mex{ribTab} is a partition filled with
  ribbons of length $k$ (for detail on ribbons see
  \Mup{combinat::ribbons}), and each ribbon is labelled with a non
  negative integer. The labels of each ribbon are placed on the
  diagonals of equations $y=x+pk$ with $p$ integer. \\ For a given
  ribbon, the tail is the lower right cell, the head is the upper left
  one and its evaluation is its label.\\ We also defined a
  semi-standard filling for a ribbon tableaux in this way: the head of
  a ribbon \Mex{R} must be at the right of a ribbon labelled with an
  integer less or equal than the evaluation of \Mex{R}, and the tail
  must be upon a ribbon labelled with an integer strictly less than
  the evaluation of \Mex{R}. \\A ribbon tableau is said standard if
  all its ribbons are labelled with different integers from 1 to the
  numbers of ribbons in the tableau (if $k=1$ ribbon tableaux are
  usual tableaux, see \Mup{combinat::tableaux}). \\Here is an example
  of a 3-ribbon tableau of shape \Mex{[5,4,3,2,1]} and evaluation
  \Mex{[3,1,1]}:}

    \begin{Mexout}
                                    +---+
                                    |   |
                                    +   +---+
                                    | 3     |
                                    +---+---+---+
                                    |         2 |
                                    +---+---+---+---+
                                    |   | 1     |   |
                                    +   +---+   +   +---+
                                    | 1     |   | 1     |
                                    +---+---+---+---+---+
\end{Mexout}
 \eng{The expression representation of a ribbon tableau is the list of its rows where the label of each ribbon is moved to its head position and zero everywhere else. For the previous example, the expression is \Mex{[[3], [0,0], [2,0,0], [1,1,0,1],[0,0,0,0,0]]}. }


    \eng{We define a statistic, the spin, for a ribbon tableau as follows: we add the height minus 1 of each ribbon and divide the result by two. \\And we define the costatistic as the cospin of a ribbon tableau of a given shape and evaluation by computing the max of the spins of all ribbon tableaux of the same shape and evaluation minus its spin (this statistic is always a positive integer). For the previous example, the spin is 2.}
    \eng{With these two statistics we can define two generating polynomials of the set of all ribbon tableaux of a given shape, evaluation and length, they are called the spin and cospin polynomials.}

\end{MDetails}

%

\begin{MExample}{ex-tab}
  \eng{There are $155$ standard tableaux of shape the ribbon  $[3, 2, 2]$:}
  \begin{Mexin}
    combinat::ribbonsTableaux::count([6,6,6,6,6,6],[3,3,3,3],3)
  \end{Mexin}
  \begin{Mexout}
                              236
  \end{Mexout}
\end{MExample}

%

\begin{MExample}{tab-word}
  \eng{Finding the image of a given ribbon tableaux by the Stanton White bijection:}
  \begin{Mexin}
    combinat::ribbonsTableaux::StantonWhite(combinat::ribbonsTableaux
    ([[4,0,0,4,4,0],[3,3,0,0,0,0],[0,0,0,3,0,0],
                            [2,0,0,2,2,0],[1,1,0,0,0,0],[0,0,0,1,0,0]]));
  \end{Mexin}
   \begin{Mexout}
        --     -- +---+---+  +---+---+  +---+---+ -- --
        |      |  | 4 | 4 |  | 3 | 4 |  | 3 | 3 |  |  |
        |      |  +---+---+  +---+---+  +---+---+  |  |
        |  [], |  | 2 | 2 |, | 1 | 2 |, | 1 | 1 |  |  |
        --     -- +---+---+  +---+---+  +---+---+ -- --
  \end{Mexout}
\end{MExample}
%


%
\begin{MExample}{spinTableauAndPolynomial}
\eng{The first function computes the spin of a given tableau and the second one computes both the spin and the cospin polynomials for a given shape, evaluation and length}
 \begin{Mexin}
   combinat::ribbonsTableaux::spinTableau(combinat::ribbonsTableaux(
         [[4,0,0,4,4,0],[3,3,0,0,0,0],[0,0,0,3,0,0],
             [2,0,0,2,2,0],[1,1,0,0,0,0],[0,0,0,1,0,0]]));
 \end{Mexin}
 \begin{Mexout}
                               4
 \end{Mexout}
\begin{Mexin}
  combinat::ribbonsTableaux::spinAndCospinPolynom([6,6,6,6,6,6],[3,3,3,3],3);
\end{Mexin}
\begin{Mexout}
    12       11       10       9       8       7       6
[4 q   + 11 q   + 29 q   + 47 q  + 56 q  + 47 q  + 29 q  +

       5      4     8       7       6       5       4
   11 q  + 2 q , 2 q  + 11 q  + 29 q  + 47 q  + 56 q  +

       3       2
   47 q  + 29 q  + 11 q + 4]
\end{Mexout}
\end{MExample}

%

 \begin{MBackground}
  \eng{Ribbon tableaux were first introduced by Stanton and White for counting colored permutations. With spin or cospin polynomials we can defined two $q$-analogues of product of Schur functions. The specialization of these functions for some values of $q$ provide conjectures for plethystic substitutions. }
\end{MBackground}

\MSuperDomain{Dom::BaseDomain}

\begin{MAxioms}
  \Mup{Ax::canonicalRep}
\end{MAxioms}

\begin{MChanges}
  \version{3.1}%
  \eng[New library]{\Mthis{} is a new library}%
\end{MChanges}

\end{Mhelppage}





