Combinatorics

Combinatorics

There are various combinatorial numbers available, for example the Stirling numbers of the second kind which you can obtain using
stirling_second_number()
Their table, and also the table of the Stirling numbers of the first kind, can be evaluated by
stirling_first_tafel(), stirling_second_tafel().
These routines provide the upper left hand corner of prescribed number of rows and columns. The Kostka numbers are of course there:
kostka_number()
as well as their table:
kostka_tafel().
Moreover, there are, as it was already mentioned, several tables and values of characters available, for symmetric groups, alternating groups and wreath products of symmetric groups, see the appropriate documentation files. This part of the program system is of course in rapid progress.

An important part of the combinatorics used in the representation theory of symmetric groups can be formulated in terms of tableaux.

Example: Here is a routine that allows to construct all the tableaux of prescribed content (cf. ex19.c):
```...
scan(PARTITION,a); println(a);
scan(PARTITION,b); println(b);
kostka\_tab(a,b,c);println(c);
...
```
Here is the output which you obtain according to the input of the partitions 23 and 11111 of 5:

23

11111

4 5

1 2 3

3 5

1 2 4

3 4

1 2 5

2 5

1 3 4

2 4

1 3 5

Please note that the standard Young tableaux are the tableaux of content 11..., and they are written in French notation, and their order is the lexicographic order or the word obtained by reading the entries of the lowest row from left to right first, and then the entries of the next row, and so on.

There are also routines that give this row word or the column word of skew tableaux.

harald.fripertinger@kfunigraz.ac.at,
last changed: November 19, 2001

 Combinatorics