Suppose we have a sequence of distinct $a_i$'s: $\left\{ a_1, a_2, \ldots, a_n \right\}$. We also have a sequence of not necessarily distinct $b_i$'s: $\left\{ b_1, b_2, \ldots, b_n \right\}$. Each $b_i$ corresponds to how many $a_i$'s there are. For example, if $a_1 = 2$ and $b_1 = 3$, then there are $3$ $2$'s. The function I have in mind, denoted $\tau_j$, is a sum of all possible combinations of products of the $a_i$'s where we have all possible combinations such that the sum of the powers of each term equals $j$.
I provide the following example. Suppose we have $a_1 = a_2 = a_3$ and $a_4 = a_5$. Let's rewrite/rename this so we just have two distinct $a_i$'s: $a_1, a_2$. We can see that $b_1 = 3$ and $b_2 = 2$. Therefore:
\begin{equation}
\tau_0 \left( a_1, a_2 \right) = 1
\end{equation}
\begin{equation}
\tau_1 \left( a_1, a_2 \right) = a_1 + a_2
\end{equation}
\begin{equation}
\tau_2 \left( a_1, a_2 \right) = a_1^2 + a_1 a_2 + a_2^2
\end{equation}
\begin{equation}
\tau_3 \left( a_1, a_2 \right) = a_1^3 + a_1^2 a_2 + a_1 a_2^2
\end{equation}
\begin{equation}
\tau_4 \left( a_1, a_2 \right) = a_1^3 a_2 + a_1^2 a_2^2
\end{equation}
\begin{equation}
\tau_5 \left( a_1, a_2 \right) = a_1^3 a_2^2
\end{equation}
As we can see, the sum of the powers in each term in each $\tau_j$ is equal to $j$. Furthermore, any power of any $a_i$ cannot exceed its associated $b_i$. I am thinking for each term in the sum of each $\tau_j$, we have to cycle through powers of each $a_i$ from $0$ to $b_i$ under the condition that the sum of the powers must equal $j$.
But the question is: how do we write that? Can we have a sum symbol under a sum symbol, as a condition (by "under", I mean where we would typically have "$i=0$")? What is the proper notation for this? Any assistance is greatly appreciated.
Best Answer
You could use multi-index notation viz.$$\tau_j=\sum_{|\alpha|=j\land\alpha\le b}a^\alpha.$$Here $a^\alpha:=\prod_ia_i^{\alpha_i}$, $|\alpha|:=\sum_i\alpha_i$ (a multi-index $\alpha$ is required to satisfy $\alpha_i\ge0$), and $\alpha\le b$ abbreviates $\alpha_i\le b_i$ for all $i$.