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Excuse me for this canonical or simple question. But someone, please, can explain me, also with simplest example for a teacher of an high school, the significance of these symbols/operators?

$$\sum_{\text{cyc}}, \qquad \prod_{\text{cyc}}, \qquad \color{red}{?}$$

PS: I require the simplest explanation.

J. W. Tanner
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Sebastiano
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    In what context have you seen these symbols? I'm inclined to think that these are sums and products over elements in cyclic groups, or permutation groups potentially. – Osama Ghani Dec 26 '20 at 23:16
  • @OsamaGhani Hi, in some questions of algebra-precalculus where there are inequalities: https://math.stackexchange.com/questions/1775572/olympiad-inequality-sum-limits-cyc-fracx48x35y3-geqslant-fracxy – Sebastiano Dec 26 '20 at 23:18
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    $\sum\limits_{cyc} a^3+b^2+c=(a^3+b^2+c)+(b^3+c^2+a)+(c^3+a^2+b)$, for example – J. W. Tanner Dec 26 '20 at 23:20
  • @J.W.Tanner Please, can you delete your comment and can you give me an example for sum and the prod? Thank you very much. – Sebastiano Dec 26 '20 at 23:56
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    Sebastiano--- Tanner's comment is already an example of cyclic sum. To get one for cyclic product just change to $\Pi a^3b^2c$ (with cyc under the big Pi). – coffeemath Dec 27 '20 at 00:00
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    @coffeemath Thank you also for you. But..I wanted give also an upvote and a check mark$\ddot\smile$. I like give the upvotes. – Sebastiano Dec 27 '20 at 10:52

1 Answers1

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A cyclic summation cycles through all the variables.

For example, if there are three variables, $a, b,$ and $c$,

then $\sum\limits_{cyc} (a^3+b^2c)$ is the sum of $a^3+b^2c$ and that expression with the variables cycled through

(i.e., $a\mapsto b, b\mapsto c, c\mapsto a$, and also $a\mapsto c, b\mapsto a, c\mapsto b$):

$(a^3+b^2c)+(b^3+c^2a)+(c^3+a^2b)$.

Similarly, the cyclic product $\prod\limits_{cyc}(a^3+b^2c)=(a^3+b^2c)(b^3+c^2a)(c^3+a^2b).$

J. W. Tanner
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