## Tableaux

Tableaux are objects with two components,
- the
*shape*
(Umriß) and
- the tableaux itself, which is a particular
matrix.

The entries of the matrices are INTEGERobjects or
empty objects. Look at the following example.
**Example: **
The tableau
3

2 3

1 2 3

is the MATRIXobject

[1:2:3]

[2:3:#]

[3:#:#]

and the shape is the PARTITION 123.

But you can also have
TABLEAUX whose shape is of type SKEWPARTITION. Please note the following:
**SYMMETRICA uses the French notation, which means that partitions are increasing
sequences. The tableaux are placed in the first quadrant of the plane, and so the
longest row is the lowest one, and since we use the standard coordinates, this
longest row is the first row of the tableaux, although it was entered last in
the
partition!**

**Example: **
Here is an example:
...
scan(TABLEAUX,a);
println(a);
...

If you have the corresponding program compiled, then, after entering *a.out* you will first
be asked if its shape is a proper partition or a skew partition, you have to reply that question either
by *P* or by *S.* If you intend to enter a partition, then you will first be asked for its length, and
then for the partition, as you know. After that you will be asked for the entries of the first row.
Recall the above quotation: If you want the shape to be the partition 2 3 of 5, say, then that first
row will be the row of length 3.
For example, if you want to enter the above tableau, you will successively have to enter the
following lines:

P

3

1 2 3

1 2 3

2 3

3

For further information you should also compare the file
*kostka.doc.*

harald.fripertinger@kfunigraz.ac.at,

last changed: November 19, 2001