Kostka numbers

Let $\lambda$ a sequence of non increasing integers.
It can be seen as the size of a jagged array

Let $\mu$ a sequence of $\sum {\lambda }_i$consecutive integers starting with 1.

We fill in the array with the elements of $\mu$ using the following rules:
- in each row the elements must be in ascending order (equality permitted)
- in each column, the element are strictly in acending order

We defined $K(\lambda ,\mu )$ as the number of such arrays.
It is called the Kostka number

If $\lambda =\{3,2\}$ and $\mu =\{1,2,3,3,4\}$, there are 3 possible arrays:

Therefore $K(\{3,2\},\{1,2,3,3,4\})=3$

What is $K(\{6,4,4,3\},\{1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9\}$?

[My timing: 4 sec ]