Question

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?

Answer 1

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.