Consider the Schur polynomial $s_\lambda(x_1,\dotsc,x_k)$. It is easy to see from the hook-content formula for counting the number of semi-standard tableaux, that the function $$ n \to s_{n \lambda}(1,1,\dotsc,1) = |SSYT(n \lambda,k)| $$ is a polynomial in $n$ with non-negative coefficients. Here, $SSYT(\lambda,k)$ denotes the set of semi-standard tableaux of shape $\lambda$, and no entry exceeding $k$. The notation $n\lambda$ denotes element-wise multiplication.

Now, what about the map $$n \to s_{n \lambda/n\mu}(1,1,\dotsc,1) = |SSYT(n \lambda/n\mu,k)|$$ where we now consider skew Schur polynomials?

It is easy to prove that this is again a polynomial (it is the Ehrhart polynomial of certain Gelfand-Tsetlin polytopes), but there is a notorious lack of a hook-content formula for counting skew SSYTs.

**Is there some argument that proves that the above map is a polynomial with non-negative coefficients?**