FUNCTION: ListTab - generate all tableaux of a given shape and evaluation

CALLING SEQUENCE:

ListTab(part, eval)

TAB[ListTab](part, eval)

PARAMETERS:

part = any list denoting a partition

eval = any list denoting an evaluation

SYNOPSIS:

• The ListTab function generates all tableaux of a given shape part and of a given evaluation eval. Note that this vector eval possibly has some 0.

• When called with a third parameter, say 'nb', it returns the number of such tableaux.

• Using a third parameter 'mintab'=list_of_lists, one specifies that the entries of the tableaux are componentwise bigger or equal to those of 'mintab'. For instance, 'mintab'=[[6], [-infinity, 2]] as well as 'mintab'=[[6], [1,2]], gives tableaux [..., [x, ...], [y, z, ...]] with x>=6, y>=1, z>=2.

• One can use both 'nb' and 'mintab' options as third or fourth argument at the same time.

• The generator is more efficient when eval is sorted as an increasing sequence of integers. It can be useful if you solely want the number of tableaux because sorting the evaluation does not change this number.

• Whenever there is a conflict between the function name ListTab and another name used in the same session, use the long form TAB['ListTab'].

EXAMPLES:

> with(TAB):
> l:=ListTab([3,2], [0,1,0,0,1,2,0,0,1]);
 
   l := [[[6, 9], [2, 5, 6]], [[5, 9], [2, 6, 6]], [[6, 6], [2, 5, 9]]]
 
> ListTab([3,2],  [0,1,0,0,1,2,0,0,1], 'mintab'=[[6],[-infinity]]);
 
                  [[[6, 9], [2, 5, 6]], [[6, 6], [2, 5, 9]]]
 
> map(Tab2Mat, l);
 
                    [ 6  9    ]  [ 5  9    ]  [ 6  6    ]
                   [[         ], [         ], [         ]]
                    [ 2  5  6 ]  [ 2  6  6 ]  [ 2  5  9 ]
 
> ListTab([5,5], [1$10], 'nb');
 
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