% $Date: 2003/09/23 20:48:30 $ $Author: nthiery $ $Revision: 1.1.1.2 $

\begin{Mhelppage}{Part2ListHook}{V. Prosper}%
  \begin{MInfo}
    \eng{calculates the list of hook lengths}
  \end{MInfo}
  
  \begin{MCalls}
  \Mcall{part}
\end{MCalls}

\begin{MParameters}
  \param{part}
  \eng{any list denoting a partition} 
\end{MParameters}

\begin{MIntro}
  \eng{

The {\Mthis} function computes the list of all hook lengths of a
given  partition.
The result is sorted into non-increasing order.
 

The arm and leg length of a cell \Mmath{(i,j)}{(i,j)} are respectively
\Mmath{part[i] - j}{\Mex{part}_i-j} and \Mmath{cpart[j] - i}{\Mex{cpart}_j-i}
where \Mex{cpart} is the conjugate of \Mex{part}.
 

The call \Mex{{\Mthis(part,a)}} computes the Jack-hook-product.
\Mex{a} may be any expression.
This is the product
\Mmath{a*arm_length[i,j] + leg_length[i,j] + 1}
      {\Mex{a}\times\mathrm{arm\_length}(i,j)+\mathrm{leg\_length}(i,j)+1}
over all cells of the diagram of the partition part.
 

The call \Mex{{\Mthis}(part,q,t)} computes the two-variable hook
polynomial.
The arguments \Mex{q} and \Mex{t} can be any expressions.
This is the product
\Mmath{1 - q^arm_length[i,j] x t^(1+leg_length[i,j])}
    {1-q^{\mathrm{arm\_length}(i,j)}\times t^{1+\mathrm{leg\_length}(i,j)}}
over all cells of the diagram of the partition part.
 
}
\end{MIntro}

\begin{MExample}{ex1}
\begin{Mexin}
muEC::PART::Part2ListHook( [3,1] );
\end{Mexin}
\begin{Mexout}
                         [4, 2, 1, 1]
\end{Mexout}
\begin{Mexin}
4! / _mult( op(%) );
\end{Mexin}
\begin{Mexout}
                               3
\end{Mexout}
\begin{Mexin}
muEC::PART::Part2ListHook( [2,1], z );
\end{Mexin}
\begin{Mexout}
                             z + 2
\end{Mexout}
\begin{Mexin}
muEC::PART::Part2ListHook( [2,1], z, t );
\end{Mexin}
\begin{Mexout}
                         2               2
                     - (t  z - 1) (t - 1)
\end{Mexout}
\end{MExample}

\MRelatedFunctions{Part2Conjugate,Part2Border,Part2Diagonal,Part2Mat,ListPart,TYP::IsPart}
\end{Mhelppage}
