# Sum notation $\sum_{\sigma\in\{\pm 1\}^n}$?

I would like to know what the following sum notation means:

$$\sum_{\sigma\in\{\pm 1\}^n}\left(\prod_{1\leq i\leq n}F(x_i^{\sigma_i})\right)$$

where $n$ is a positive integer, $x_i$ are some variables and $F$ are functions. In the above only the notation under the sum $\sigma\in\{\pm 1\}^n$ is puzzling to me. What does it mean?

EDIT:

I guess a bit of context might accelerate things. I found the notation in the following definition:

EDIT2:

Yves Daoust seems to suggest that $\{\pm1\}^n$ means collections of ordered lists of length $n$ with all possible combinations of $+1$ or $-1$, so that i.e. for $n=3$ we have:

$$\{+1,+1,+1\}\\\{-1,+1,+1\}\\\{+1,-1,+1\}\\\{+1,+1,-1\}\\\{+1,-1,-1\}\\\{-1,+1,-1\}\\\{-1,-1,+1\}\\\{-1,-1,-1\}$$

and each $\sigma$ denotes a full list, while $\sigma_i$ refers to elements from a list. Then the sum is over all $2^n$ different lists.

• $\{\pm1\}^n$ probably denotes the set of the sequences of $n$ values $+1$ or $-1$, and the sum carries on them (there are $2^n$ such sequences). – Yves Daoust Apr 5 '16 at 20:44
• @YvesDaoust I have added an edit2 to the question. Did I understand your idea correctly? – Kagaratsch Apr 5 '16 at 20:51
• Yep, but I wouldn't use braces to denote the lists, but parenthesis. Braces are for sets. – Yves Daoust Apr 5 '16 at 20:58