<s>[INST] Simplify summation over collection Consider a collection $V_t$ of vectors of dimension $n$ created as follows:
When $n= 3, t=0$
$$V_0=\left\{\begin{pmatrix}0\\0\\0\end{pmatrix}\right\}$$
When $n=3, t=1$
$$V_1=\left\{\begin{pmatrix}1\\0\\0\end{pmatrix},\begin{pmatrix}0\\1\\0\end{pmatrix},\begin{pmatrix}0\\0\\1\end{pmatrix}\right\}$$
When $n=3,t=2$
$$V_2=\left\{\begin{pmatrix}2\\0\\0\end{pmatrix},\begin{pmatrix}0\\2\\0\end{pmatrix},\begin{pmatrix}0\\0\\2\end{pmatrix},\begin{pmatrix}1\\1\\0\end{pmatrix},\begin{pmatrix}1\\0\\1\end{pmatrix},\begin{pmatrix}1\\1\\0\end{pmatrix},\begin{pmatrix}0\\1\\1\end{pmatrix},\begin{pmatrix}1\\0\\1\end{pmatrix},\begin{pmatrix}0\\1\\1\end{pmatrix}\right\}$$
Basically every the sum of every element within a vector is equal to $t$ and to create $V_{t+1} $ you have to loop through every element of every vector of $V_{t}$ and create a new vector with that element incremented by $1$.
Now consider a sub-collection $F_t$ of $V_t$ that just contains the vectors where every element is at least $1$, evidently when $n=3$, $F_0,F_1,F_2=\emptyset$.
I am trying to find a simpler expression for: $$\sum_{r=\text{Element of }F_t}\prod_{k=1}^nx_k^{r_{k}}$$
for example if $n=2,t=3$:
$$V_3=\left\{\begin{pmatrix}3\\0\end{pmatrix},\begin{pmatrix}0\\3\end{pmatrix},\begin{pmatrix}2\\1\end{pmatrix},\begin{pmatrix}1\\2\end{pmatrix},\begin{pmatrix}2\\1\end{pmatrix},\begin{pmatrix}1\\2\end{pmatrix},\begin{pmatrix}2\\1\end{pmatrix},\begin{pmatrix}1\\2\end{pmatrix}\right\}$$
$$\implies F_3=\left\{\begin{pmatrix}2\\1\end{pmatrix},\begin{pmatrix}1\\2\end{pmatrix},\begin{pmatrix}2\\1\end{pmatrix},\begin{pmatrix}1\\2\end{pmatrix},\begin{pmatrix}2\\1\end{pmatrix},\begin{pmatrix}1\\2\end{pmatrix}\right\}$$
$$\implies \sum_{r=\text{Element of }F_3}\prod_{k=1}^2x_k^{r_{k}}= x_1^2x_2^1 + x_1^1x_2^2 + x_1^2x_2^1 + x_1^1x_2^2+x_1^2x_2^1+x_1^1x_2^2=3x_1^1x_2^2+3x_1^2x_2^1$$
I tried doing it using an expression similar to the multinomial summation however this didn't work since it's not taking into count repeated elements. However I imagine if there is a way to write it with a sum it'll look similar.
 [/INST] You can find a simple expression for this using the principle of inclusion exclusion to subtract the monomials where some variable has an exponent of zero.
$n=2$: $$(x_1+x_2)^t-x_1^t-x_2^t$$
$n=3$: $$(x_1+x_2+x_3)^t-(x_1+x_2)^t-(x_1+x_3)^t-(x_2+x_3)^t+x_1^t+x_2^t+x_3^t$$
etc. In general, the expression for $n$ is
$$
\sum_{S\subseteq \{1,\dots,n\}}(-1)^{n-|S|} \left(\sum_{i\in S}x_i\right)^t
$$
</s>