Cycle (Permutation)
A Subset of a Permutation whose elements trade places with one another. A cycle decomposition of aPermutation can therefore be viewed as a Class of a Permutation Group. For example, in thePermutation Group 
, 
is a 3-cycle (
, 
, and 
) and 
is a 1-cycle (
). Every Permutation Group on 
symbols can be uniquely expressed as a product ofdisjoint cycles. The cyclic decomposition of a Permutation can be computed in Mathematica
(WolframResearch, Champaign, IL) with the function ToCycles and the Permutation corresponding to a cyclicdecomposition can be computed with FromCycles. According to Vardi (1991), the Mathematica code for ToCyclesis one of the most obscure ever written.
To find the number 
of 
cycles in a Permutation Group of order , take
where
is the Stirling Number of the First Kind.See also Golomb-Dickman Constant, Permutation, Permutation Group, Subset
References
Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 20, 1990. Vardi, I. Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, p. 223, 1991.
- Kampyle of Eudoxus
- KAM Theorem
- Kanizsa Triangle
- Kantorovich Inequality
- Kaplan-Yorke Conjecture
- Kaplan-Yorke Dimension
- Kaplan-Yorke Map
- Kappa Curve
- Kaprekar Number
- Kaprekar Routine
- Kaps-Rentrop Methods
- Kapteyn Series
- Karatsuba Multiplication
- k-ary Divisor
- Katona's Problem
- Kauffman Polynomial F
- Kauffman Polynomial X
- k-Chain
- k-Circuit
- k-Coloring
- Kei
- Keith Number
- Keller's Conjecture
- Kelvin Functions
- Kelvin's Conjecture