Is there a explicit formula for the number of Semi-standard Young Tableaux over $\{1,\dots,n\}$ for a given partition $\lambda$ and a given type $\mu$

Question

I was given an exercise to give all SSYT over $\{1,\dots,12\}$ of shape $\lambda=(4,4,3,1)$ and type $\mu=(4,2,2,2,2,0,\dots,0)$. Now I was wondering if there is an formula to say something about the number of SSYT in general.

Answer 1

These are called Kostka numbers.

According to Wikipedia:

In general, there are no nice formulas known for the Kostka numbers. However, some special cases are known. For example, if $\mu = (1, 1, 1, ..., 1)$ is the partition whose parts are all $1$ then a semistandard Young tableau of weight $\mu$ is a standard Young tableau; the number of standard Young tableaux of a given shape $\lambda$ is given by the hook-length formula.

Wikipedia doesn't provide a reference. But according to this question it seems like Stanley says something to this effect in EC2. I recommend following that link and also the link in the comments there to MO.

I also found a relevant paper that shows that computing Kostka numbers is #P-complete.