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	✅	✅	✅
degree	Return the degree of the permutation	✅	✅	✅
descents	Return the positions of the permutation descents	✅	❌	✅
describe	Return a table describing the permutation	✅	✅	✅
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	✅	❌	✅
lehmer_code	Return the Lehmer code of the permutation	✅	❌	✅
lexicographic_rank	Return the lexicographic rank of the permutation	✅	❌	✅
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	✅	✅	✅
records	Return the positions of the permutation records	✅	❌	✅
sgn	Return the sign of the permutation	✅	✅	✅
support	Return the support of the permutation	✅	✅	✅