# The Cali Garmo

does Math

## 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. $\left\{1,1,2,2\right\}$.