Trouble with understanding the definition of cyclic sums?

This is from my book:

Given an $n-$ variable expression $f(x_1, x_2, x_3, ..., x_n)$, we will denote the cyclic sum by

$\displaystyle \sum_{\sigma} f(x_1, x_2, x_3, ..., x_n) =$

$f(x_1, x_2, x_3, ..., x_n) + f(x_1, x_2, x_3, ..., x_n) + \cdots + f(x_1, x_2, x_3, ..., x_n)$.

For example, if our variables are $x, y,$ and $z$, then

$$\sum_{\sigma} x^3 = x^3 + y^3 + z^3$$ and

$$\sum_{\sigma} xz^2 = xz^2+yx^2+zy^2$$

I don't understand their examples. In the definition, they did not say anything along the lines of "if our variables are ..."; the variables just depended on what was given in the expression.

The way it seems to me, $x^3$ is $f(x)$, an expression of just one variable so according to the definition, $\sum_{\sigma} x^3 = x^3$

Similarly, $xz^2$ is $f(x, z)$ a two variable expression so $\sum_{\sigma} xz^2 =f(x, z) + f(z, x) = xz^2 + zx^2$.

I would also like to double check that I understood the original definition correctly. Is the following correct?

$\displaystyle \sum_{\sigma} f(x_1, x_2, x_3, x_4, x_5) = f(x_1, x_2, x_3, x_4, x_5) + f(x_2, x_3, x_4, x_5, x_1) + f(x_3, x_4, x_5, x_1, x_2,)+ f(x_4, x_5, x_1, x_2, x_3) + f(x_5, x_1, x_2, x_3, x_4)$

• You're right that in your examples it is not made clear what the set of variables and their ordering is. In general, this has to be given, typically indexed variables, ordered by rising indices, or variables with different letters, then in alphabetic order. You're also right that, if no such indication is given, the "natural" one is taken as I just described. So if, in the function terms, $x$ and $z$ occur, the "natural" guess would be that your set is $\{x,y,z\}$. If only $x$ occurs, you would have no idea. In any case, failure of giving the set of variables and their ordering is bad style. – Andreas Jun 9 '17 at 15:18
• @Andreas Thanks. Is there a way to embed the set into the notation, or is the set usually defined separately. For example, can I write $\displaystyle \sum_{\sigma \in \{x, y, z, w \}} x^2+y$? – Ovi Jun 9 '17 at 20:40
• In your typesetting, it might be unclear that it is cyclic. I would write $\sum_{cyc \{x,y,w,z \}}$ which indicates the set, the initial ordering and the instruction to consider all cyclic permutations. – Andreas Jun 19 '17 at 9:18
• @Andreas Okay thanks. The $\sigma$ notation is just used in the current book I am reading. Perhaps it might be better if we write $(x, y, w, z)$ instead of $\{x, y, w, z \}$? – Ovi Jun 19 '17 at 13:14

Let $f(x,y,z) = xz^2$. Then $f(y,z,x) = yx^2$ and $f(z,x,y) = zy^2$. Similarly, if $f(x,y,z)=x^3$, then $f(y,z,x)=y^3$ and $f(z,x,y)=z^3$.