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The number of standard tableaux of a given shape is given by the
formula
Related to the Macdonald polynomials one has the q,tcoefficients and referred to as the q,tKostka polynomials.
If q and t are both set to 1, we have that
and
are both equal to the number of standard tableaux of shape lambda.
Conjecture:The polynomials (and ) have nonnegative integer coefficients andare a generating function for the number of standard tableaux of shape lambda. 
This result (at least the postitivity part) is known for
when it is of the form (2^{a}1^{b}), (32^{a}1^{b}), (42^{a}1^{b}), (a21^{b}), (a1^{b}) when it is of the form (a1^{b}), (a21^{b}), (a31^{b}) OR the conjugate partition of any of these results. 
Note that if q=0 in the polynomial , we have the standard single variable Kostka polynomials which are known to tcount the column strict tableaux of shape and content .
Below are the tables for the transformed q,tKostka polynomials for n=2,3 and 4. You may also find these tables up to n=7 in PDF/poscript and LaTeX form on this web site. and Maple programs by Stembridge and GarsiaTesler.


