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. .