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Let $\lambda$ be a partition and define $T_\lambda^n(k)$ to be the number of semi-standard young tableaux of shape $(k\lambda_1,k\lambda_2,\dots,k \lambda_n).$ Now, one can prove that $T_\lambda^n(k)$ is a polynomial in $k.$

Is this well-known? Is there a known formula for $T_\lambda^n(k)$? Would it be interesting to find a general formula for this polynomial?

(I know about the hook-length formula, but since it is a product with $|\lambda|$ factors, this is not really useful for studying asymptotics of $T_\lambda^n(k)$.)

(The asymptotics for the corresponding Schur polynomials are also quite interesting. The roots of $S_{k \lambda}(t,1,1,\dots,1) = 0$ will for example accumulate close to the unit circle as $k$ grows.)

EDIT: It seems that finding Kostka numbers, $K_{k\lambda, k\beta}$ is a refinement of the above problem, and it is known that these grows polynomially. An article from 2007, Degrees of Stretched Kostka coefficients, by B. McAllister finds the degree of these, but the exact formulas for the polynomials seems to be unknown.

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up vote 7 down vote accepted

If my memory serves me correctly, the number of semi-standard Young tableaux of shape $\lambda$ and with entries in $[n]$ (I suppose you mean that) can be interpreted as the number of lattice points in a certain polytope determined by $n$ and $\lambda$, and scaling up the parts of $\lambda$ is tantamount to refining the lattice. If this is true, it would easily give you the polynomiality statement (and the degree of the polynomial). Since the number is also the dimension of a certain representation, you also have Weyl's dimension formula.

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Your memory is correct; you are thinking of the Gelfand-Tsetlin polytope. –  David Speyer Aug 31 '12 at 15:49
    
Yes, I found some articles now that I knew what to google for. Added some info I gathered. Is it true that the polynomiality of the streched Kostka coefficient is not straightforward? (I mean, the proof involves these polytopes in an essential way?) –  Per Alexandersson Aug 31 '12 at 16:14
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For new exact formulas for these quantities, and their asymptotics, you maybe should see the recent papers

http://arxiv.org/abs/1109.1412

http://arxiv.org/abs/1208.3443 (Theorem 1.2)

and especially http://arxiv.org/abs/1301.0634 (Theorem 1.1 and 1.2)

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