Notation for expressing a product over combinations of indices in expansion of $\prod^m_{n=1} (1 + a_n)$?

Question

What is a formal way of notating the expansion of the following product?

$$p_m = \prod^m_{n=1} (1 + a_n).$$

Alternatively, what is the formal way of notating a product operator over combinations of indices?

Context.

Taking $m = 3$, the expansion is

$$\begin{aligned} p_3 &= (1 + a_1)(1 + a_2)(1+ a_3) \\ &= 1 + a_1 + a_2 + a_3 + a_1a_2 + a_2a_3 + a_1a_3 + a_1a_2a_3 \end{aligned}$$

With a view to finding an expression for $p_m$ I tried using a sum-product to get something a bit like

$$p_3 = 1 + \sum^3_{n=1} \prod^n_{i = ???}(a_i).$$

But I am unsure how to express the fact that I only want a product over combinations of indices. However I can certainly recall seeing probability and statistics papers using a clean notation to express this idea. However, I can't remember what they are and Googling for "product operator over combinations of indices" doesn't yield fruit. I would appreciate some assistance on this.

Answer 1

I would simply write $$ \prod_{n=1}^m (1+a_n) = \sum_{S\subseteq \{1,2,\dots,m\}} \prod_{n\in S} a_n. $$ But if you wanted to collect the terms corresponding to subsets of a particular size, I would write $$ \prod_{n=1}^m (1+a_n) = 1 + \sum_{k=1}^m \sum_{\substack{S\subseteq \{1,2,\dots,m\} \\ \#S = k}} \prod_{n\in S} a_n. $$

Answer 2

A similar question was asked here and answered by Moh. Using the notation established therein:

For any set $S$ and any natural number $n \geq 0$, let $\binom{S}{n}$ denote the set of all $n$-length combinations on $S$. For example, $$\binom{\{a, b, c\}}{2} = \{\{a, b\}, \{a, c\}, \{b, c\}\}$$

Adapting this, define the set $A = \{a_1, \dots, a_m \}$, and so your product can be expressed as

$$p_m = \prod^m_{n=1}(1 + a_n) = 1 + \sum^m_{n=1}\sum_{A' \in \binom{A}{m}} \prod_{a \in A'} a.$$

Where the inner sum is over all sets $A'$ of $m$-length combinations in $\binom{A}{m}$ and the product is taken over all elements $a$ in a set $A'$.