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Kostka Numbers

By Cali G , Published on Mon 13 January 2020
Category: math / symmetric functions

Kostka Numbers

Let n be a natural number. Recall that the Young diagram of a partition \lambda of n is the diagram which has \lambda_1 number of boxes in row 1, \lambda_2 boxes in row 2, 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 K_{\lambda,\mu} is the number of semi-standard Young tableau of shape \lambda with content \mu.

As an example, let \lambda = (3,2) and \mu = (2,1,1,1). Then (3,2) implies we have the following Young diagram: and since \mu = (2,1,1,1) then we must fill it in with the number \left\{1,1,2,3,4\right\}.

In this case we have 3 different fillings:

Therefore, the Kostka number is 3:

Skew Kostka Numbers

We can define Kostka numbers in exactly the same way for skew shape semi-standard Young tableau.

As an example let \lambda = (3,2), \beta = (1) and \nu = (2,2). Then the skey shape is given by \lambda / \beta = (3,2)/ (1):

Then K_{\lambda/\beta,\nu} = 2 since there are only two ways to fill the above shape with content (2,2), i.e. \left\{1,1,2,2\right\}.