Mathematica 9 is now available
THIS IS DOCUMENTATION FOR AN OBSOLETE PRODUCT.
SEE THE DOCUMENTATION CENTER FOR THE LATEST INFORMATION.
Wolfram Mathematica Documentation Center
DOCUMENTATION CENTER SEARCH
New to Mathematica? Find your learning path »
Mathematica > Mathematics and Algorithms > Discrete Mathematics > Permutations > Cycles >
BUILT-IN MATHEMATICA SYMBOLTutorials »|See Also »|More About »

Cycles

Cycles
represents a permutation with disjoint cycles .
MORE INFORMATION
  • The cycles of a permutation are given as lists of positive integers, representing the points of the domain in which the permutation acts.
  • A cycle represents the mapping of the to . The last point is mapped to .
  • Points not included in any cycle are assumed to be mapped onto themselves.
  • Cycles must be disjoint, that is, they must have no common points.
  • Cycles objects are automatically canonicalized by dropping empty and singleton cycles, rotating each cycle so that the smallest point appears first, and ordering cycles by the first point.
  • Cycles represents the identity permutation.
EXAMPLESCLOSE ALL
Basic Examples   (2)
A permutation with two cycles:
Automatic evaluation to a canonical form:
A permutation with two cycles:
In[1]:=
Click for copyable input
Out[1]=
 
Automatic evaluation to a canonical form:
In[1]:=
Click for copyable input
Out[1]=
Scope   (2)
Permutations can involve any positive integers, with cycles of any length:
Identity permutation:
Properties & Relations   (9)
The identity permutation contains no cycles in its canonical form:
Permutation applied to a single point:
Points not present in the cycles are mapped onto themselves:
Cycles given in SparseArray form are automatically converted into normal lists:
Generate the list of permutations corresponding to a symmetric group:
Permutations are numerically ordered by comparing their respective lists of images:
Canonical Mathematica ordering of Cycles objects:
The identity is always sorted first:
A way to compute the inverse of a permutation:
Possible Issues   (3)
Only positive integers can appear in cycles:
All integers must be distinct:
Permutation objects with symbolic arguments return unevaluated:
Neat Examples   (1)
Graph representation of a permutation:
The inverse permutation has the arrows reversed:
New in 8
Ask a question about this page  |  Suggest an improvement  |  Leave a message for the team
Format:   HTML  |  CDF