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?

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    $\begingroup$ Would it be more precise to say "nλ denotes the partition where each part of λ is multiplied by n" instead of "The notation nλ denotes element-wise multiplication"? $\endgroup$ – Wolfgang Dec 8 '18 at 21:37
  • $\begingroup$ @Wolfgang Sure, that is another way to phrase it. $\endgroup$ – Per Alexandersson Dec 8 '18 at 21:58
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    $\begingroup$ To make sure I understand, $k$ is fixed throughout, yes? $\endgroup$ – David E Speyer Dec 9 '18 at 15:25
  • $\begingroup$ @DavidESpeyer: Yes, exactly. $\endgroup$ – Per Alexandersson Dec 9 '18 at 15:32

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