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.
- Hexomino
- Hex (Polyhex)
- Hex Pyramidal Number
- Heyting Algebra
- H-Fractal
- H-Function
- Hh Function
- Higher Arithmetic
- Highest Weight Theorem
- HighLife
- Highly Abundant Number
- Highly Composite Number
- Higman-Sims Group
- Hilbert Basis Theorem
- Hilbert Curve
- Hilbert Function
- Hilbert Hotel
- Hilbert Matrix
- Hilbert Number
- Hilbert Polynomial
- Hilbert's Axioms
- Hilbert-Schmidt Norm
- Hilbert-Schmidt Theory
- Hilbert's Constants
- Hilbert's Inequality