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.
- Relative Minimum
- Relaxation Methods
- Remainder
- Remainder Theorem
- Rembs' Surfaces
- Removable Crossing
- Removable Singularity
- Rencontres Number
- Rendezvous Values
- Repartition
- Repdigit
- Repeating Decimal
- Repfigit Number
- Replicate
- Representation
- Reptend Prime
- Rep-Tile
- Repunit
- Residual
- Residual vs. Predictor Plot
- Residue Class
- Residue (Complex Analysis)
- Residue (Congruence)
- Residue Index
- Residue Theorem (Complex Analysis)