Difference between revisions of "Cyclic sum"

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(Rigorous definition)
 
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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,a_3,\ldots a_n)</math>. The cyclic sum <math>\sum_{cyc} f(a_1,a_2,a_3,\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>
 
<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>

Latest revision as of 17:39, 4 August 2012

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 $f(a_1,a_2,a_3,\ldots a_n)$. The cyclic sum $\sum_{cyc} f(a_1,a_2,a_3,\ldots a_n)$ is equal to

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

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

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