Let $f(x_1,\dots,x_k)$ be a multivariate polynomial of degree $m$. It is a seemingly well known fact that $$ g(n) = \sum_{1 \leq i_1 < \dots < i_k \leq n} f(i_1,\dots,i_k) $$ is a polynomial of degree $m+k$, though I can't find an explicit reference. I am interested in finding a reference for this fact and the stronger statement that $g$ is divisible by $\binom{n}{k}$ (note this is the evaluation when $f=1$). The univariate case is discussed in this question.
Example: Let $f(x_1,x_2) = x_1^2 x_2$. Then $$ g(4) = 1^22 +1^23 + 2^23+1^24+2^24+3^24 = 73. $$ Note $g(4) - g(3) = 4 (1^2 + 2^2 + 3^2)$, and more generally $$ g(n+1) - g(n) = n \sum_{i=1}^{n-1} i^2 = n^2(n+1)(2n+1)/6. $$ Since the difference is a polynomial of degree $4$, we see $g(n)$ must be a polynomial of degree 5. To compute it, we use polynomial interpolation with the values $$ g(-1) = g(0) = g(1) = 0, g(2) = 2, g(3) = 17, g(4)=73 $$ and $g(-1) = g(0) - (-1) \cdot (-2+1) \cdot (-1+1) = 0$. This gives $$ g(n) = \frac{1}{120} n(n-1)(n+1)(8n^2 +5n-2) = \binom{n}{2} \frac{(n+1)(8n^2+5n-2)}{60}. $$
More generally, one can prove this result for monomials by induction following the outline of our example, and the rest follows by linearity. The divisibility by $\binom{n}{k}$ follows since $0,1,\dots,k-1$ are roots of $g$.
I would like to understand the result in a more conceptual way. Is there a general principle from which the polynomiality and divisibility by $\binom{n}{k}$ are obvious? Is there a combinatorial interpretation for the remaining term after dividing by $\binom{n}{k}$?
