Difference between revisions of "Cyclic sum"
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==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>. 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== | ||
Revision as of 16:40, 11 February 2008
| This is an AoPSWiki Word of the Week for February 4-11 |
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
![\[f(a_1,a_2,a_3,\ldots a_n)+f(a_2,a_3,a_4,\ldots a_n,a_1)+f(a_3,a_4,\ldots a_n,a_1,a_2)\ldots+f(a_n,a_1,a_2,\ldots a_{n-1})\]](http://latex.artofproblemsolving.com/9/4/f/94f2282a9eaced21157c0de211a1267cabb983f1.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.
