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?


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

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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:


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.

  • 2
    $\begingroup$ $\{\pm1\}^n$ 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). $\endgroup$ – Yves Daoust Apr 5 '16 at 20:44
  • $\begingroup$ @YvesDaoust I have added an edit2 to the question. Did I understand your idea correctly? $\endgroup$ – Kagaratsch Apr 5 '16 at 20:51
  • 2
    $\begingroup$ Yep, but I wouldn't use braces to denote the lists, but parenthesis. Braces are for sets. $\endgroup$ – Yves Daoust Apr 5 '16 at 20:58

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