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.
- Solid Angle
- Solid Geometry
- Solid of Revolution
- Solid Partition
- Solidus
- Solitary Number
- Soliton
- Solomon's Seal Knot
- Solomon's Seal Lines
- Solomon's Seal Polygon
- Solvable Congruence
- Solvable Group
- Solvable Lie Group
- Soma Cube
- Somer-Lucas Pseudoprime
- Sommerfeld's Formula
- Somos Sequence
- Sondat's Theorem
- Sonine Polynomial
- Sonine-Schafheitlin Formula
- Sonine's Integral
- Sophie Germain Prime
- Sorites Paradox
- Sorting
- Sort-Then-Add Sequence