Difference between revisions of "Cyclic sum"

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==Rigorous definition==
 
==Rigorous definition==
Consider a function <math>f(a_1,a_2,a_3,\ldots a_n)</math>. The cyclic sum <math>\sum f(a_1,a_2,a_3,\ldots a_n)</math> is equal to  
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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,a_3,\ldots a_n)+f(a_2,a_3,a_4,\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>
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<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==
If a summation is specified without additional arguments, it is generally assumed to be a cyclic sum. A cyclic sum can also be 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.
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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
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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==
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*[[Symmetric sum]]
 
*[[Symmetric sum]]
  
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[[Category:Algebra]]
 
[[Category:Definition]]
 
[[Category:Definition]]

Revision as of 09:11, 28 April 2022

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 $f(a_1, a_2, \ldots, a_n)$. The cyclic sum $\sum_{cyc} f(a_1, a_2, \ldots, a_n)$ is equal to

\[f(a_1, a_2, \ldots, a_n) + f(a_2, a_3, \ldots, a_n, a_1) + f(a_3, a_4, \ldots, a_n, a_1, a_2) + \ldots + f(a_n, a_1, a_2, \ldots, a_{n-1}).\]

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: $\sum_{a,b,c}\frac{ab}{cd}.$ Note that a cyclic sum need not cycle through all of the variables.

A cyclic sum is also sometimes specified by $\sum_{cyc}$. This notation implies that all variables are cycled through.

See also