# Difference between revisions of "Cyclic sum"

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− | Consider a function <math>f(a_1,a_2,a_3,\ldots a_n)</math>. The cyclic sum <math>\ | + | 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> |

## 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 . The cyclic sum is equal to

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.