Difference between revisions of "Cyclic sum"
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A '''cyclic''' sum is a [[summation]] that cycles through all the values of a function and takes their sum, so to speak. | A '''cyclic''' sum is a [[summation]] that cycles through all the values of a function and takes their sum, so to speak. | ||
==Rigorous definition== | ==Rigorous definition== | ||
| − | Consider a function <math>f(a_1,a_2 | + | Consider a function <math>f(a_1, a_2, \ldots, a_n)</math>. The cyclic sum <math>\sum_{cyc} f(a_1, a_2, \ldots, a_n)</math> is equal to |
| − | <cmath>f(a_1,a_2 | + | <cmath>f(a_1, a_2, \ldots, a_n) + f(a_2, a_3, \ldots, a_n, a_1) + f(a_3, a_4, \ldots, a_n, a_1, a_2) + \ldots + f(a_n, a_1, a_2, \ldots, a_{n-1}).</cmath> |
Note that not all permutations of the variables are used; they are just cycled through. | Note that not all permutations of the variables are used; they are just cycled through. | ||
==Notation== | ==Notation== | ||
| − | A cyclic sum is often specified by having the variables to cycle through underneath the sigma, as follows: <math>\sum_{a,b,c}\frac{ab}{cd}</math> | + | A cyclic sum is often specified by having the variables to cycle through underneath the sigma, as follows: <math>\sum_{a,b,c}\frac{ab}{cd}.</math> Note that a cyclic sum need not cycle through all of the variables. |
| − | A cyclic sum is also sometimes specified by <math>\sum_{cyc}</math>. | + | A cyclic sum is also sometimes specified by <math>\sum_{cyc}</math>. This notation implies that all variables are cycled through. |
==See also== | ==See also== | ||
*[[Summation]] | *[[Summation]] | ||
*[[Symmetric sum]] | *[[Symmetric sum]] | ||
| − | + | *[[PaperMath’s sum]] | |
| − | [[Category: | + | [[Category:Algebra]] |
[[Category:Definition]] | [[Category:Definition]] | ||
Latest revision as of 11:51, 8 October 2023
A cyclic sum is a summation that cycles through all the values of a function and takes their sum, so to speak.
Rigorous definition
Consider a function 
. The cyclic sum 
is equal to
![\[f(a_1, a_2, \ldots, a_n) + f(a_2, a_3, \ldots, a_n, a_1) + f(a_3, a_4, \ldots, a_n, a_1, a_2) + \ldots + f(a_n, a_1, a_2, \ldots, a_{n-1}).\]](http://latex.artofproblemsolving.com/e/9/f/e9f2db4d8c2f1ebf235f73f55367ac9eb256a8e2.png)
Note that not all permutations of the variables are used; they are just cycled through.
Notation
A cyclic sum is often specified by having the variables to cycle through underneath the sigma, as follows: 
Note that a cyclic sum need not cycle through all of the variables.
A cyclic sum is also sometimes specified by 
. This notation implies that all variables are cycled through.
