Kostka Numbers
Let be a natural number. Recall that the Young diagram of a partition
of
is the diagram which has
number of boxes in row
,
boxes in row
, etc. Recall further that the Young tableau is the Young diagram filled in with some numbers and that the tableau is semi-standard if the rows are weakly increasing and the columns are strongly increasing.
Definition: The Kostka number is the number of semi-standard Young tableau of shape
with content
.
As an example, let and
. Then
implies we have the following Young diagram:
and since
then we must fill it in with the number
.
In this case we have different fillings:
Therefore, the Kostka number is :
Skew Kostka Numbers
We can define Kostka numbers in exactly the same way for skew shape semi-standard Young tableau.
As an example let ,
and
. Then the skey shape is given by
:
Then since there are only two ways to fill the above shape with content
, i.e.
.