The Hook lenght formula gives the number of standard Young tableaux on a given diagram. 


A variant gives the number of semistandard tableuax. 


Does there exist a formula for counting "weighted tableaux"? By weighted tableaux I mean that there exists a vector $(a\_1,\dots,a\_n)$ and I only want to count the tableaux with entries in $\{1,\dots,n\}$ such that $i$ appears exactly $a\_i$ times.


Does anybody have a reference for this?



Dimension of spaces of invariants/tableaux functions
The numbers you refer to are known as Kostka numbers. They are discussed in standard references like Fulton's Young Tableaux and Stanley's Enumerative Combinatorics. The weights of a tableaux are often referred to as their content, as well.


