Elements

Elements#

The symmetric group boasts a multitude of elements, precisely \(n!\), where \(n\) is the number of elements on which the group operates. These elements can be represented in various forms, such as permutations, cycles, or cycle decompositions.

Symmetria furnishes three distinct classes – Permutation, Cycle, and CycleDecomposition – to represent an element within the symmetric group in different way. Up to you to pick the representation which fit at best your needs.

Note

The class Permutation is, in some sense, the basic class to represent a permutation. It is raccomanded to work with it if you don’t have specific needs to have cycles or cycles decomposition of permutations.

The following table summarize the functionalities provided by each class. Here, P denotes the class Permutation, C the class Cycle, and CD the class CycleDecomposition.

overview#

Feature

Description

P

C

CD

__call__

Call the permutation on an object

✅

✅

✅

__mul__

Multiplication (composition) between permutations

✅

❌

✅

__pow__

Power of a permutation

✅

✅

✅

ascents

Return the positions of the permutation ascents

✅

❌

✅

cycle_decomposition

Cycle decomposition of the permutation

✅

✅

✅

cycle_type

Return the cycle type of the permutation

✅

❌

✅

cycle_notation

Return the cycle notation of the permutation

✅

✅

✅

descents

Return the positions of the permutation descents

✅

❌

✅

exceedances

Return the positions of the permutation exceedances

✅

❌

✅

inverse

Compute the inverse of the permutation

✅

✅

✅

inversions

Return the inversions of the permutation

✅

✅

✅

is_conjugate

Check if two permutations are conjugate

✅

❌

✅

is_derangement

Check if the permutation is a derangement

✅

✅

✅

is_even

Check if the permutation is even

✅

✅

✅

is_odd

Check if the permutation is odd

✅

✅

✅

is_regular

Check if the permutation is regular

✅

❌

✅

map

Return the map defining the permutation

✅

✅

✅

one_line_notation

Return the one line notation of the permutation

✅

❌

❌

orbit

Compute image of a given element under the permutation

✅

✅

✅

order

Return the order of the permutation

✅

✅

✅

sgn

Return the sign of the permutation

✅

✅

✅

support

Return the support of the permutation

✅

✅

✅