PermutationPower

The $n$th power of a permutation under composition; a negative $n$ powers the inverse.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguagePermutationPower✓↗

PermutationPower(p, n)$p^n$, in the same notation as p

SageMathenumeratio_permutation_power(p, n)

Typed signatures

Type	Package	Applies
(expression<Cycles> | list<integer> | permutation, integer) -> expression<Cycles> | list<integer> | permutation	combinatoricsreplaces groupalgebra	always
(expression<Cycles> | list<integer>, integer) -> expression<Cycles> | list<integer>	groupalgebra	always

Domain: Permutations

Details

• Either notation or a Permutation, as PermutationProduct reads them; the result keeps the notation it was given
• $p^0$ is the identity, and $p^n$ is the identity whenever PermutationOrder divides $n$
• A non-integer exponent leaves the call unevaluated

Examples

Basic #

live ↗

In$\mathrm{PermutationPower}(\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack\bigr\rbrack), 2)$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 3, 2\bigr\rbrack\bigr\rbrack)$

Known value · wolframscript 15.0

Scope #

the inverse

live ↗

In$\mathrm{PermutationPower}(\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack\bigr\rbrack), -1)$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 3, 2\bigr\rbrack\bigr\rbrack)$

Known value · wolframscript 15.0

live ↗

In$\mathrm{PermutationPower}(\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack\bigr\rbrack), -2)$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack\bigr\rbrack)$

Known value · wolframscript 15.0

live ↗

In$\mathrm{PermutationPower}(\bigl\lbrack2, 3, 1\bigr\rbrack, 2)$Out$\bigl\lbrack3, 1, 2\bigr\rbrack$

Known value · wolframscript 15.0

a Permutation gives a Permutation

live ↗

In$\mathrm{PermutationPower}({\operatorname{Permutation}}([2, 3, 1]), 2)$Out${\operatorname{Permutation}}([3, 1, 2])$

Known value · wolframscript 15.0, on the one-line word

Properties #

the identity

live ↗

In$\mathrm{PermutationPower}(\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack\bigr\rbrack), 0)$Out$\mathrm{Cycles}(\bigl\lbrack\bigr\rbrack)$

Known value · wolframscript 15.0

$7 \equiv 3 \pmod 4$, and the 4-cycle has order 4

live ↗

In$\mathrm{PermutationPower}(\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3, 4\bigr\rbrack\bigr\rbrack), 7)$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 4, 3, 2\bigr\rbrack\bigr\rbrack)$

Known value · wolframscript 15.0

About

Examples	7
Checked against	Wolfram Language 7 agreeSageMath 7 agree
Crosswalk	Wolfram LanguageSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts

See also: PermutationProduct, InversePermutation, PermutationOrder