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	✅	❌	✅
cycle_decomposition	Cycle decomposition of the permutation	✅	✅	✅
cycle_notation	Return the cycle notation of the permutation	✅	✅	✅
is_derangement	Check if the permutation is a derangement	✅	✅	✅
map	Return the map defining the permutation	✅	✅	✅
one_line_notation	Return the one line notation of the permutation	✅	❌	❌
support	Return the support of the permutation	✅	✅	✅
orbit	Compute image of a given element under the permutation	✅	✅	✅
order	Return the order of the permutation	✅	✅	✅