Let $k,n\in\mathbb{N}^+$, $\vec{x}=\langle x_1,\cdots,x_n\rangle$ be an $n$-tuple, and let $e_k(\vec{x})$ be the elementary symmetric polynomial of degree $k$ over $n$ variables (clearly with $k\le n$). For fixed $k$, I'm wondering what can be said about the image of $e_k$ over $(\mathbb{N}^+)^n$; that is, the output over all $n$-tuples of positive integers. Positivity is needed because it's trivial if we allow non-positive arguments: if any of the $x_i$ are zero, then it reduces to a lower degree, and if negative integers are allowed, then we are able to generate $\mathbb{Z}$ by Bezout's Theorem.
Clearly $e_k(\vec{x})$ will have $\binom{n}{k}$ terms, so that is the minimum value with each $x_i=1$. However, playing around in Mathematica suggests there is a threshold $M(k,n)$ such that $e_k(\vec{x})\ne M(k,n)$ for any $\vec{x}$, but there for every $m>M(k,n)$ there is some $n$-tuple $\vec{v}$ with $e_k(\vec{v})=m$. In other words, $e_k$ is surjective except for finitely many values, and I'm wondering what the last of these values is.
For example, with $k=2$ and $n=4$, it seems (although I can't quite prove it) every value except $\{1,2,3,4,5,7,8,10,11,14,16,19,20,26,31,34,40,55\}$ can be reached, implying $M(2,4)=55$. I tried to play around with several of the forms, in particular looking at reside classes for $e_2(x,a,b,1)$ for $b=1,2$ and values of $a$. This is reminiscent of finding the Frobenius number of a set, but perhaps using the symmetry of the polynomials it's possible to say something like, "eventually you can eliminate all residue classes of a certain modulus, so it remains to check just the first few cases."