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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?

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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.

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  • $\begingroup$ What is the number for the partition $2n=(2,2,2,...,2)$ and $a_i=2$ for all $i$ ? $\endgroup$
    – Ram
    Commented Feb 12, 2015 at 6:11
  • $\begingroup$ sorry, just ignore the last comment. I need the number for the partition $2n=(n,n)$ and $a_i=2$ for all $i=1,2, \cdots ,n$. $\endgroup$
    – Ram
    Commented Feb 12, 2015 at 6:23

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