# Cycles

Cycles[{cyc1,cyc2,}]

represents a permutation with disjoint cycles cyci.

# Details • The cycles cyci of a permutation are given as lists of positive integers, representing the points of the domain in which the permutation acts.
• A cycle {p1,p2,,pn} represents the mapping of the pi to pi+1. The last point pn is mapped to p1.
• 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.

# Examples

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## Basic Examples(2)

A permutation with two cycles:

Automatic evaluation to a canonical form:

## 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 Wolfram Language 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: