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.